12/7/12 HW00: Introduction to MasteringPhysics 1/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« HW00: Introduction to MasteringPhysics Due: 11:59pm on Wednesday, August 29, 2012 Note: To understand how points are awarded, read your instructor's Grading Policy. A message from your instructor... The purpose of the following exercises is to familiarize you with the system you will be using for the rest of your course. These exercises are not intended to teach or test your knowledge of any specific subject material. Therefore, you will not be penalized for using hints or submitting incorrect answers. Welcome! Mastering presents homework items assigned by your instructor and works with you to answer them. Homework items typically have an introduction, possibly figures, and one or more parts for you to answer. Type of help offered Mastering tells you immediately whether or not your answers are correct. Usually, you will have multiple chances to arrive at the correct answer. Your instructor will determine how many tries you have available. In many items, hints are available to help you if you get stuck. If you don't need the hints to solve the problem, you can still use them for review later on. If you submit an incorrect answer, Mastering often responds with specific, helpful feedback. Mastering is forgiving of many typos and formatting mistakes. If it can't figure out what you entered, it will let you know and give you another chance. These exercises were chosen specifically to lead you through the key features of Mastering and are not intended to test your knowledge of any specific subject material. Therefore, on this item you will not be penalized for using hints and submitting incorrect answers. In fact, you should submit incorrect answers and use the hints to see what happens! Part A How many squares are in this grid ? Note that the figure link lets you know that a figure goes along with this part. This figure is available to the left. Enter your answer as a number in the box below and then submit your answer by cl icking Submit. ANSWER: Correct What you are reading now is called a "follow-up comment." These comments typically offer more information or provide an interpretation of the answer you just obtained. Before you move on to a slightly more challenging question, have a quick look at the other buttons available around the answer box. Give up gives you the answer to the question if you can't figure it out on your own. Your instructor controls whether or not this button is available. My Answers brings up a new window that lists all of the answers you have submitted for this question, along with any helpful feedback you received for incorrect submissions. Grading Number of squares = 5 12/7/12 HW00: Introduction to MasteringPhysics 2/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« See the help file available by clicking the Help tab in the top right corner, if you want to know more about how grading works. Here is the most important information you'll need. In a graded homework item, each part counts equally toward your score on the overall item. If you get full credit on each part, you will receive full credit for the problem. You may lose a fraction of the credit for a part when you submit an incorrect answer. Whether you do lose credit and how much you lose are set by your instructor. However, you won't lose credit for most types of formatting mistakes or for submitting a blank answer. As you might expect, you will receive no credit if you use the Show Answer button. If you just can't figure out a question, there is a way to get partial credit by using hints, as the following part will illustrate. Part B What is the magic number? Note that there is a figure also associated with this part. However, the figure for Part A may still be visible on the left. To view the figure associated with Part B, click on the figure link. A new figure should appear on the left. You could try to guess the magic number but you would probably use up all your tries before getting the answer. Notice the new Hints button underneath the answer box for this question. Clicking this button w il l open up a l ist of hints that w il l guide you to the correct number. Hint 1. Different types of hints and their impact on grading Notice that there are three hints for this question. You are not required to use all of the hints or to use them in order. Each hint has a tagline that describes its contents. Based on the tagline you can decide whether or not a particular hint will be useful to you. There are two kinds of hints. Some hints, such as Hint 2 below, just provide you with information. Other hints, such as Hint 3 below, give you an opportunity to answer a simpler question that is related to the main question you are solving. These hints either have questions in the tagline or tell you to do something (e.g., Find..., Determine..., Identify..., etc.). There are two ways that this type of hint can help you: Answering the simpler question gives you a chance to check that you are on the right track. If you correctly answer the simpler question, you will receive partial credit for the part even if you are unable to answer the main question. Your instructor may choose to give you a bonus for not using hints or to deduct a small penalty for using hints. If you are stuck, using the hints will usually result in a higher score than simply trying to guess because you may lose fewer points for opening a hint than for getting the answer to the main question incorrect. There is a more detailed explanation of how hints are graded in the help available by clicking the Help tab in the top right corner of your screen in the main Mastering window. In this problem, however, you will not lose any credit for using the hints. Now, open up the second hint for some help finding the magic number. Hint 2. How to approach the problem Although you could try to guess the magic number you would most likely exhaust your tries before getting the correct answer. To help you, the magic number is , where is a number between 1 and 10. Hint 3. What is ? Recall that the previous hint stated that the magic number is , where is a number between 1 and 10. Specifically is an even number between 1 and 10. Try to guess the value of . You may submit as many guesses as you need. Enter each guess into the answer box that fol lows. ANSWER: Correct Now that you have determined , compute to find the magic number. = 4 ANSWER: Correct Your instructor may choose to give you a bonus for not using hints or to deduct a small penalty for using hints. If you are stuck, using the hints will usually result in a higher score than simply trying to guess because you may lose fewer points for opening a hint than for getting the answer to the main question incorrect. Note that you are never required to use the hints; if you want to figure the question out on your own, go ahead! Notice that a new button, Review Part, appears when you correctly answer a part with hints. This button allows you to review all of the hints for that part, even if you didn't need them to get the answer. This is a useful way to review the question when studying for a test. You do not lose any credit for reviewing the hints after you have answered the question. If you didn't look at all of the hints while answering the last question, you should read through them now for some important information about hints and hint grading. Part C Multiple-choice questions have a special grading rule determined by your instructor. Assume that your instructor has decided to grade these questions in the following way: If you submit an incorrect answer to a multiple-choice question with options, you will lose of the credit for that question. Just like the similar multiple-choice penalty on most standardized tests, this rule is necessary to prevent random guessing. If a multiple-choice question has five answer choices and you submit one wrong answer before getting the question correct, how much credit will you lose for that part of the question? ANSWER: Correct Your instructor may choose not to deduct of the credit for a multiple-choice question with options. To see how your instructor is grading you, click your instructor's Grading Policy on your assignment page. If you click on the Continue button before finishing all the Parts, you will see a message reminding you that you need to complete each Part to get credit. If you have completed the item, clicking Continue will take you to the next item on the Assignment. At any time you may click on the Provide Feedback link to access a survey page without losing your work. Once you have completed an item, you may access your score from the assignment. Your score will display below the item title. Introduction to Numeric Answers This exercise is not intended to test your knowledge of any specific subject material. Therefore, on this problem you will not be penalized for using hints or submitting incorrect answers. Overview When an assignment question requires that you enter a numeric answer, you will see an answer box like the one illustrated here. magic number = 60 100 50 33 25 20 12/7/12 HW00: Introduction to MasteringPhysics 4/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« To answer this type of question, you will need to type the numeric value for distance in the rectangle between the quantity you are solving for (distance) and the units (meters). In addition to typing alphanumeric values from your keyboard, you can enter non-numeric information by using either the menu tabs above the rectangular answer box (for more information click Help) or the equivalent key strokes, as listed in the Shortcuts menu. Grading Tolerance For grading, your numeric answers often (but not always) can be within a tolerance range of the official answer. For example, if the answer to a numeric problem with a tolerance of 2% were 105, answers between 103 and 107 would be graded as correct. If you submit an answer that is rounded to within the grading tolerance but is not the exact answer, you will receive full credit. If this value is needed in future parts use the full precision value for subsequent parts. Significant figures Most questions with numeric answers will require your answer to be given to at least three digits or significant figures. Your answer may be graded as incorrect if you have calculated correctly but then rounded your final answer to too few digits. If a different number of significant figures is required, this will be part of the question's answer instructions. When you need to do multiple calculations to get an answer, use more significant figures than required in each calculation and round at the end only. Rounding too early can cause your final answer to be outside the tolerance range. Part A For most answers, you will simply enter your numeric answer directly into the space provided to the right of the equal sign. Answer the following question by typing the numeric answer into the answer box. If you have a gross of items, you have 144 items. If you buy a gross of eggs, how many dozen eggs do you have? Express your answer in dozens. Do not enter the units; they are provided to the right of the answer box. Hint 1. How many items are in a dozen? If you have a dozen items, how many items do you have? ANSWER: ANSWER: Correct Part B When entering large numbers in the answer box, do not use commas. For example, enter 1276400 for the number 1,276,400. Do not enter 1,276,400. If you accidentally enter commas, you will get a message that your answer has the wrong number of terms. Answer the following question by typing the numeric answer into the answer box. What is the sum of 9260 and 3240? 6 items 12 items 13 items number of eggs = 12 12/7/12 HW00: Introduction to MasteringPhysics 5/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« Express your answer numerically to at least three significant figures. Hint 1. How to compute the sum Recall that the sum of two numbers is the result you obtain by adding the two numbers together. ANSWER: Correct Note that you can always review exactly what you typed in the answer box by clicking My Answers. Scientific notation You may want to enter 367,000,000 in scientific notation as . There are two ways to do this. To use a template, click the button found under the menu. To use the keyboard, Instead of the symbol, use a multiplication dot ( ) by typing the keyboard multiplication symbol * (Shift + 8). To input the exponent for numbers written in scientific notation, do either of the following: Click the button found under the menu. Type ^ (Shift + 6) from your keyboard. For example, can be entered in the answer box by typing 3.67*10^8. Part C Practice entering numbers in scientific notation by entering the diameter of a hydrogen atom in its ground state, , into the answer box. Express the diameter of a ground-state hydrogen atom in meters using scientific notation. Do not enter the units; they are provided to the right of the answer box. ANSWER: Correct In some computer programming languages and software, a shorthand scientific notation for a number such as would be 3.0E12. This notation should be avoided when using the math answer box, because the E will be interpreted as a variable. To correctly enter as an answer, follow the advice given above. Part D If you are asked to provide a set of two or more numeric answers, separate them with commas. For example, to provide the year that Sputnik (the first satellite to be sent into orbit around the Earth) was launched and the year humans first walked on the Moon, you would enter 1957,1969 in the answer box. A rectangle has a length of 5.50 and a width of 12.0 . What are the perimeter and area of this rectangle? Enter the perimeter and area numerically separated by a comma. The perimeter should be given in meters and the area in square meters. Do not enter the units; they are provided to the right of the answer box. sum = 12500 = 1.06×10 −10 12/7/12 HW00: Introduction to MasteringPhysics 6/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« Hint 1. How to find the perimeter The perimeter of a two-dimensional shape is the distance around the outside edge of the shape. In the case of a rectangle, there are two sides of length and two sides of width . Therefore, if you were to walk around the outside of a rectangle, you would walk a distance of or . Hint 2. How to find the area The area of a rectangle is given by the product of its length and width : . ANSWER: Correct Introduction to Symbolic Answers This exercise is not intended to test your knowledge of any specific subject material. Therefore, on this problem you will not be penalized for submitting incorrect answers. Overview The type of answer box illustrated here lets you build a symbolic expression just as it would look in your textbook or as you might write it by hand. You use the same answer box and menu tabs to enter numeric and symbolic answers. To insert a formatting template, such as an exponent, square root, or fraction, click and then select the appropriate template. To insert lowercase Greek letters and (hbar), click and then select the letter you want. To insert uppercase Greek letters and (EMF), click and then select the letter you want. To undo your work, click . To redo your work, click . To clear your work (restart), click . To view a l ist of keyboard shortcuts, click . Here are some of the most common shortcuts: Subscript: Type _ (an underscore). Exponent: Type ^ (Shift + 6). Fraction: Type / (a forward slash). Square root: Type \sqrt. Greek letters: Type a backslash (\) and the name of the Greek character. For example, to display , you would type \delta. For uppercase Greek letters, begin the name with a capital letter. For example, you would type \Delta to display . For more information, click ( ). To identify the purpose of any icon, simply place your cursor over it. For instance, here is an example showing the Greek letter Omega: . Grading Your answers are graded according to the standard order-of-operations conventions for evaluating mathematical expressions, as follows: 1. Perform any calculations inside parentheses. perimeter, area = 35.0 , 66.0 , HW00: Introduction to MasteringPhysics 7/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« 2. Perform all multiplications and divisions, working from left to right. 3. Perform all additions and subtractions, working from left to right. For example, in the expression , first you should multiply by and then add to the total. In other words, the correct expression is . Part A Similar to what you see in your textbook, you can generally omit the multiplication symbol as you answer questions online, except when the symbol is needed to make your meaning clear. For example, is not the same as . When you need to be explicit, type * (Shift + 8) to insert the multiplication operator. You will see a multiplication dot ( ) appear in the answer box. Do not use the symbol . For example, for the expression , typing m a would be correct, but mxa would be incorrect. Enter the expression . ANSWER: Correct When entering algebraic expressions, such as ma, you can enter it using explicit multiplication m a or implied multiplication ma. Both will be accepted as correct. Part B Enter the expression , where is the lowercase Greek letter theta. ANSWER: Correct To identify a variable displayed in a specific part, place your cursor over it. For instance, here is an example showing the Greek letter theta used in this problem: . Part C Enter the expression , where is the inverse sine function. ANSWER: Correct Use the same notation to enter other inverse trigonometric functions, for example and for the inverse cosine and tangent functions respectively. Part D Enter the expression , where is the uppercase Greek letter Delta. = = = 12/7/12 HW00: Introduction to MasteringPhysics 8/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« ANSWER: Correct Part E Enter the expression , where is N-naught (an N with a subscript zero) and is the lowercase Greek letter lambda. ANSWER: Correct Introduction to Sorting Questions This exercise is not intended to test your knowledge of any specific subject material. Therefore, on this question you will not be penalized for using the hint or submitting incorrect answers. Overview 1. Sorting questions require you to place objects into different categories or bins. There will always be at least two bins and each bin is labeled according to the category it represents. 2. An object can only belong to one bin. 3. A bin can remain empty. 4. Before an answer is submitted, it is possible to rearrange the objects by dragging them to new locations. To start over, click Reset. 5. Once you're satisfied with your sorting, click Submit. Part A Correctly classify the given food items as either a fruit or a vegetable. If you need help, look at the hint available by clicking Hints. Drag the foods into the appropiate bins. Fruits should be placed in the left bin. Vegetables should be placed in the right bin. Hint 1. How to distinguish fruits from vegetables Botanically speaking, anything that grows from a flower and bears seeds is considered a fruit. For example, oranges, grapes, and strawberries are all fruits. True vegetables are those that come from a part of the plant that is not the flower, such as leaves, stems, and roots. Thus lettuce, cabbage, and radishes are all vegetables. ANSWER: = = 12/7/12 HW00: Introduction to MasteringPhysics 9/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« Correct A tomato comes from a flower and contains seeds, so it is a fruit and is classified as a berry. An avocado contains a large seed in its center similar to peaches, apricots, and mangos. Plants such as cucumbers, squash, green beans, peppers, and eggplants are also fruits by this definition, although they are usually used as vegetables in cooking. Keep in mind that you can refer to this question or consult the Help if you have difficulty using this answer type later on. Introduction to Ranking Questions This exercise is not intended to test your knowledge of any specific subject material. Therefore, on this question you will not be penalized for submitting incorrect answers. Overview 1. Rankings are usually ordered from left to right. Check the problem instructions to be sure. 2. To rank two (or more) objects that are equivalent, drag one object into the ranking bin then drag the second object to overlap it. These objects will then automatically align to be slightly offset (so you can see both of them at once). You will know they are equal because a vertical gray bar will appear behind them to indicate an equivalent ranking group. 3. Sometimes, it will be impossible to rank the objects based on the information given. In such a case, click on the box labeled "The correct ranking cannot be determined" below the ranking window. 4. Before an answer is submitted, it is possible to rearrange the objects by dragging them to new locations. To start over, click Reset. 5. Once you're satisfied with your ranking, click Submit. Part A Each of these geometric shapes has a different number of sides. Arrange the shapes in order from the shape with the greatest number of sides to the shape with the fewest number of sides. Rank these shapes from greatest to fewest number of sides. To rank items as equivalent, overlap them. ANSWER: 12/7/12 HW00: Introduction to MasteringPhysics 10/64 Correct Note that when two or more objects are equivalent they are placed one atop another. They will appear slightly offset, but the background color behind them will change to light gray to indicate that the items are in the same equivalent ranking group. Keep in mind that you can refer to this question or consult the Help if you have difficulty using this answer type later on. Introduction to Graphing Questions This exercise is not intended to test your knowledge of any specific subject material. Therefore, on this question you will not be penalized for submitting incorrect answers. Overview When an assignment question requires that you answer by plotting a graph, you will see the following buttons: To add a new graph, click . You will first be prompted to select a label. Then you can add points. (After adding your first point, the tool automatically changes to add points mode.) You can create as many "unlabeled graphs" as you like -- these will not be graded. To add or edit points on a graph, select or . This button toggles between these two modes. About adding points: Your cursor appears as a crosshair with coordinates next to it. To add a point, click a location. To add multiple points at a time, click and drag your mouse. About editing a graph: Using a mouse: To move a point, click to select the point and then drag it to the new location. To move the entire graph to a new location, click between points on the graph and then drag. Using your keyboard: With a point or graph selected, press any arrow key to reposition. About selecting/deselecting points and graphs: Click on any point to select it. To select a graph, click between points. Actively selected points and graphs are highlighted. To deselect a point or graph, click an empty area of the workspace. To delete a selected point or graph, click or . To delete using your keyboard, press Delete or Backspace after selecting a point or selecting a graph. To change the graph label for the active graph, click . To reset your graphs to the most recently submitted answer (restart), click . For help with graphing, click . 12/7/12 HW00: Introduction to MasteringPhysics 11/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« Part A Create a graph of . Construct a graph corresponding to the l inear equation . Hint 1. Identify a point to graph You need to create a plot of . To identify points on this graph you must solve for the values of when takes on different values. What is when ? Express your answer numerically. ANSWER: Correct Now, to create your graph, find at least one other pair of and values to plot. ANSWER: Correct Now that you have some experience with graphing task question types you can tackle harder graphs. Introduction to Vector Drawing Questions When an assignment question requires that you answer by drawing vectors or moments, you will see the following buttons. To draw a new vector, click and select the label for the vector that you want to draw. You can draw the vector by clicking on the start point and dragging your mouse to the end point. 4 12/7/12 HW00: Introduction to MasteringPhysics 12/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« To edit an ex isting vector, click to select either the vector, its start point, or its end point. Drag the selected vector or point to a new location. Some questions will ask you to mark moments. To mark a moment, click once to add a clockwise arrow, again to add counterclockwise arrow, and a third time to return to a state of no moment/torque action (no directional arrow). To deselect a vector or moment, click on any empty area of the workspace. To view the sum of al l the vectors in your draw ing, click . This option is not available in every problem. To delete a selected vector or moment, click . To delete using your keyboard, press Delete or Backspace after selecting a vector or moment. To change a label, get information about a vector or moment, to change the length and angle of a vector using the keyboard or to change the direction of a moment, click to display attribute windows such as the following: Here you can change a label by selecting another vector or moment. Vector properties (Length and Angle) will display or the moment direction, if they are required for you to complete the drawing. To reset your vectors or moments to the most recently submitted answer (restart), click . For help with drawing vectors or moments, click . Part A Every morning Ann walks her dog through the park, shown as a green square on the diagram below. They start at point 1, walk one block up the street, take a turn at the corner labeled 2, and walk diagonally through the park to point 3. To return home, they walk two blocks down the street and turn right at the corner labeled 4. Draw the path taken by Ann as she walks her dog. Represent each segment of Anna's walk with a vector. The vectors should start and end at the centers of the red dots located on the image. ANSWER: Correct Besides drawing vectors by connecting two given dots, you will often have to draw vectors of a given length or at a specific angle. In these cases, click , then adjust the vector from the properties window to the desired length or angle. Part B Now imagine that instead of walking along the path , Ann walks 80 meters on a straight line north of east starting at point 1. Draw Ann's path. Represent Ann's walk with a vector of length 80 meters. 12/7/12 HW00: Introduction to MasteringPhysics 13/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« Draw the vector starting at point 1. The length displayed in the Vector Info properties w indow is given in meters. ANSWER: Correct You can use the button to adjust the length, and orientation of a vector: 1. Click on the button to display the properties window. 2. Drag the properties window to a location where it will not prevent you from seeing and manipulating the vector. 3. Draw the vector at the desired location. As you draw the vector, notice the information displayed in the properties window (length and angle). Depending on the question, some information may not be available in the properties window. When an angle is displayed, it is the angle, measured counterclockwise, that the vector makes with respect to the positive axis pointing toward the right (east). 4. Adjust the vector until the desired length and angle are displayed in the properties window or just manually enter them. Part C The diagram below shows a force being applied on a beam. Mark the direction of moment at the fulcrum by clicking on the dot indicated by . Click once to add a clockw ise arrow, again to add counterclockwise arrow, and a third time to return to a state of no moment/torque action (no directional arrow). ANSWER: 12/7/12 HW00: Introduction to MasteringPhysics 14/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« Correct Reviewing the Fundamentals Try this final item to review some of the key concepts you've learned. Part A You are starting a new item and after reading the first part you realize you have no idea how to go about answering it. What should you do? ANSWER: Correct The hints are designed for exactly this reason: to give you something to work with if you are absolutely stuck. You will most likely score higher (and learn more!) if you use the hints when you need them rather than guessing or giving up. Part B You have been working on an item for a while and after a few missteps you've come up with an answer. However, there is one particular thing that you're not 100 sure of. What should you do? Select al l that apply. ANSWER: Guess randomly and hope for some useful feedback. Use the available hints. Request the solution immediately. Check for any hints that address the part of the calculation you're unsure about. Return to the question after you've spoken with an instructor or classmate. Submit your answer and then adjust it according to any feedback you receive. None of the above. 12/7/12 HW00: Introduction to MasteringPhysics 15/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« Correct There is no single right way to proceed if you've made some progress on an item but are not 100 sure of your work. If you submit the answer, you might be right and there is a good chance you will get some useful feedback. There is also a good chance that the hints address whatever issue you are unsure about. Finally, if your instructor allows it, talking over your solution with someone else could be a good way to go.You should do whatever you find works best for you. Part C You've just solved a problem and the answer is the mass of an electron, . How would you enter this number into the answer box? Enter your answer in ki lograms using three significant figures. Note that the units are provided for you to the right of the answer box. Hint 1. Multiplication and superscripts You may use the scientific notation template to enter your answer, or type the following: 9.11*10^-31. The keyboard multiplication symbol * (Shift + 8) will appear as a multiplication dot, and the carat symbol ^ (Shift + 6) will superscript the line. Another way to get an exponent is by clicking in the menu. ANSWER: Correct Part D A friend in your class tells you that she never uses hints when doing her Mastering homework. She says that she finds the hints helpful, but when the hint asks another question it increases the chance that her score on the problem will go down. She feels like it isn't worth the risk.You reassure her that there is nothing to fear about opening a hint that asks a question. Which of the following are good reasons for your friend not to worry? Select al l that apply. ANSWER: Correct That's it! You're done! We hope you enjoy using Mastering. A message from your instructor... The remaining problems are for credit and to help you brush up on the required mathematics used in Physics 19. = 9.11×10 −31 As an incentive for thinking hard about the problem, your instructor may choose to apply a small hint penalty, but this penalty is the same whether the hint simply gives information or asks another question. Getting the correct answer to the question in a hint actually gives you some partial credit, even if you still can't answer the original question. The only way to lose additional partial credit on a hint is by using the "give up" button or entering incorrect answers. Leaving the question blank will not cost you any credit. None of the above. 12/7/12 HW00: Introduction to MasteringPhysics 16/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« Understanding Vector Addition Learning Goal: To learn to add vectors. In physics, many important quantities—from the simple foundations of mechanics such as position and force to very foreign ideas such as electric current densities and magnetic fields—are vectors. Simply knowing what a vector is helps understanding, but to really use the ideas of physics and predict things, you need to be able to do calculations with those vectors. The simplest operation you might need to perform on vectors is to add them. Suppose that you are swimming in a river while a friend watches from the shore. In calm water, you swim at a speed of 1.25 . The river has a current that runs at a speed of 1.00 . Note that speed is the magnitude of the velocity vector. The velocity vector tells you both how fast something is moving and in which direction it is moving. Part A If you are swimming upstream (i.e., against the current), at what speed does your friend on the shore see you moving? Express your answer in meters per second. Hint 1. How to approach the problem If you are swimming against the current in a river, then the river is slowing you down. Therefore, your speed would be . ANSWER: Correct You see that simply adding the two speeds would have you moving at 2.25 , nine times faster than you would actually move. You likely could answer the last question without thinking about vectors at all. If a person swims against a current, it slows the person down. The speeds subtract in this case, because you are not actually adding speeds. You are adding velocities, which are vectors. To add two vectors, say , think of taking one vector ( ) and putting its tail on the head of the other vector ( ). The sum of the two vectors is then the vector that begins at the tail of and ends at the head of . Part B If instead of swimming against the current you swam directly across the river (by your reckoning) at a speed of 1.00 from left to right, which figure correctly shows the velocity vector with which your friend on the shore would see you moving? Choose the correct figure. The added vectors are shown in gray; the vector representing their sum is shown in black. 0.25 ANSWER: Correct Although drawing vectors is helpful for visualizing what happens when you add vectors, it is not a convenient way to calculate precise results: Adding components is preferable. When you add two vectors, the resulting vector's components are the sums of the corresponding components of the original vectors. For instance, consider the two vectors and with components and , respectively. If you want to find the sum , then you would simply add the x components to get the resulting x component and add the y components to get the resulting y component: . For the situation of you swimming across the river from left to right, use a standard set of coordinates where the x axis is horizontal, with positive pointing to the right, and the y axis is vertical, with positive pointing upward. Part C Which of the following gives the correct components for the current velocity and the pure swimming velocity (i.e., the velocity that you would have in still water) using this coordinate system? ANSWER: Correct Part D What is the resultant velocity vector when you add your swimming velocity and the current velocity? A B C D current: , ; swimming: , current: , ; swimming: , current: , ; swimming: , current: , ; swimming: , current: , ; swimming: , 12/7/12 HW00: Introduction to MasteringPhysics 18/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« Give the x and y components in meters per second separated by a comma. ANSWER: Correct Look at figure D in Part B. It makes sense from this figure that the components of the final vector should be , because the two vectors that you are adding are parallel to the coordinate axes. You can follow along the two vectors and see what the final components should be. Adding vectors by adding their components always works, regardless of the directions of the vectors. Part E Consider the two vectors and , defined as follows: and . What is the resultant vector ? Give the x and y components of separated by a comma. ANSWER: Correct Understanding Components of Vectors Learning Goal: To understand and be able to calculate the components of vectors. You have heard vectors defined as quantities with magnitude and direction, familiar ideas also found in statements such as “three miles northeast of here.” Components, the lengths in the x and y directions of the vector, are a different way to define vectors. In this problem, you will learn about components, by considering ways that they arise in everyday life. Suppose that you needed to tell some friends how to get from point A to point B in a city. The net displacement vector from point A to point B is shown in the figure. You could tell them that to get from A to B they should go 3.606 blocks in a direction 33.69 north of east. However, these instructions would be difficult to follow, considering the buildings in the way. Part A You would more likely give your friends a number of blocks to go east and then a number of blocks to go north. What would these two numbers be? Enter the number of blocks to go east, fol lowed by the number of blocks to go north, separated by a comma. ANSWER: 1 , -1 = 1.05 , -6.48 12/7/12 HW00: Introduction to MasteringPhysics 19/64 Correct If you call east the x direction and north the y direction, then you can see that you have broken up the displacement vector into components. The two lighter colored vectors in the figure below have magnitudes equal to the components of the displacement vector in their respective directions. As drawn, you can see that these component vectors add together to give the original vector, just as going 3 blocks east, then 2 north gets you to the same place as going 3.606 blocks in the direction 33.69 north of east. Notice that the figure with the component vectors drawn in has the shape of a right triangle. You can use trigonometry to find the components of any vector. Recall that, for some angle in a right triangle, the sine of that angle, , is defined as the length of the side (O) opposite the angle divided by the length of the hypotenuse (H) of the triangle, and the cosine of the angle, , is defined as the length of the side (A) adjacent to the angle divided by the length of the hypotenuse of the triangle. In terms of these definitions, and the hypotenuse’s length , the triangle’s sides have the following lengths: for the side opposite the angle, and for the side adjacent to the angle. Part B Consider the vector with magnitude 4.00 at an angle 23.5 north of east. What is the x component of this vector? Express your answer in meters to three significant figures. ANSWER: Correct Part C 3 , 2 blocks = 3.67 12/7/12 HW00: Introduction to MasteringPhysics 20/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« Consider the vector with length 4.00 at an angle 23.5 north of east. What is the y component of this vector? Express your answer in meters to three significant figures. ANSWER: Correct You can also think about components as projections onto the coordinate axes. Consider the shadow cast by your hand if you hold it near a movie screen. As you tilt your hand closer to the horizontal, the shadow gets smaller. If you think of your hand as a vector with tail at the base of your palm and arrow at your fingertips, the shadow's height corresponds to the y component of the vector. If you were to shine another light from above, the shadow cast below your hand would correspond to the x component. Part D What is the length of the shadow cast on the vertical screen by your 10.0 hand if it is held at an angle of above horizontal? Express your answer in centimeters to three significant figures. Hint 1. Which component do you need? Since the shadow is vertical and the horizontal direction coincides with the positive x axis, the shadow of your hand would be the y component of the "hand vector." ANSWER: Correct You can also use your knowledge of right triangles to solve the problem in reverse, that is, to find the magnitude and direction of a vector from its components. If you know the two components of a two-dimensional vector, you can use the Pythagorean Theorem to find the vector’s magnitude (i.e., length) by adding the squares of the two components and then taking the square root. In Parts B and C, the two components were 3.668 and 1.595 , and the vector’s magnitude is . Part E What is the magnitude of a vector with components (15 , 8 )? Express your answer in meters. = 1.59 5 12/7/12 HW00: Introduction to MasteringPhysics 21/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« Hint 1. More about the Pythagorean Theorem Recall from geometry that the Pythagorean Theorem says , where and are the lengths of the two legs of a right triangle and is the length of the hypotenuse. You know from the previous discussion that the x and y components of a vector can be thought of as the lengths of two legs of a right triangle with the vector itself as hypotenuse. Therefore the magnitude of the vector and the two components and must satisfy the Pythagorean Theorem: . Taking the square root of both sides gives the relation ANSWER: Correct Finding the direction from the components requires a bit of trigonometry. In a right triangle, the tangent of an angle is the length of the side (O) opposite the angle divided by the length of the side (A) adjacent to the angle. Using this definition and the figure showing the right triangle, you can see that the tangent of the angle above the positive x axis is the y component of the vector (the length of side O) divided by the x component of the vector (the length of side A): . When you use this formula, remember that you are finding the angle measured counterclockwise from the positive x axis. Sometimes you will be asked for the angle with other axes. You should be able to use the same trigonometry described here, but this formula may not be quite right. Part F What is the angle above the x axis (i.e., "north of east") for a vector with components (15 , 8 )? Express your answer in degrees to three significant figures. ANSWER: Correct Trig Functions and Right Triangles Learning Goal: To use trigonometric functions to find sides and angles of right triangles. The functions sine, cosine, and tangent are called trigonometric functions (often shortened to "trig functions"). Trigonometric just means "measuring triangles." These functions are called trigonometric because they are used to find the lengths of sides or the measures of angles for right triangles. They can be used, with some effort, to find measures of any triangle, but in this problem we will focus on right triangles. Right triangles are by far the most commonly used triangles in physics, and they are particularly easy to measure. 17 28.1 12/7/12 HW00: Introduction to MasteringPhysics 22/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« The sine, cosine, and tangent functions of an acute angle in a right triangle are defined using the relative labels "opposite side" and "adjacent side" . The hypotenuse is the side opposite the right angle. As you can see from the figure, the opposite side is the side of the triangle not involved in making the angle. The side called the adjacent side is the side involved in making the angle that is not the hypotenuse. (The hypotenuse will always be one of the two sides making up the angle, because you will always look at the acute angles, not the right angle.) The sine function of an angle , written , is defined as the ratio of the length of opposite side to the length of the hypotenuse: . You can use your calculator to find the value of sine for any angle. You can then use the sine to find the length of the hypotenuse from the length of the opposite side, or vice versa, by using the fact that the previous formula may be rewritten in either of the following two forms: or . Part A Suppose that you need to get a heavy couch into the bed of a pickup truck. You know the bed of the truck is at a height of 1.00 and you need a ramp that makes an angle of 40 with the ground if you are to be able to push the couch. Use the sine function to determine how long of a board you need to use to make a ramp that just reaches the 1.00- high truck bed at a 40 angle to the ground. Express your answer in meters to three significant figures. Hint 1. Using the sine function The ramp is the hypotenuse of the right triangle in the figure, and the side of length 1.00 is opposite the 40 angle. To find the length of the hypotenuse, use the form of the sine formula. Plugging in the given values will give you the length of the hypotenuse. ANSWER: Correct The cosine function is another useful trig function. The definition of the cosine function is similar to the definition of the sine function: 1.56 12/7/12 HW00: Introduction to MasteringPhysics 23/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« . This equation can be rearranged the same way that the equation for sine was rearranged. With the cosine of an angle, you can find the length of the adjacent side from the length of the hypotenuse, or vice versa. Part B You need to set up another simple ramp using the board from Part A (i.e., a board of length 1.56 ). If the ramp must be at a 25 angle above the ground, how far back from the bed of the truck should the board touch the ground? Assume this is a different truck than the one from Part A. Express your answer in meters to three significant figures. Hint 1. Using the cosine function The ramp is the hypotenuse of the right triangle in the figure, and the distance along the ground is adjacent to the 25 angle. To find the length of the adjacent side, use the form of the cosine formula. Plugging in the given values will give you the distance along the ground. ANSWER: Correct The third frequently used trig function is the tangent function. The tangent of an angle is defined by the equation . This equation can be rearranged the same way that the equations for sine and cosine were rearranged previously. With the tangent of an angle, you can find the length of the adjacent side from the length of the opposite side or vice versa. Part C Surveyors frequently use trig functions. Suppose that you measure the angle from your position to the top of a mountain to be 2.50 . If the mountain is 1.00 higher in elevation than your position, how far away is the mountain? Express your answer in ki lometers to three significant figures. 1.41 Hint 1. Using the tangent function The height of the mountain is opposite the 2.50 angle of the right triangle in the figure, and the distance to the mountain is adjacent to the 2.50 angle. To find the distance to the mountain, use the form of the tangent formula. Plugging in the given values will give you the distance to the mountain. ANSWER: Correct All of the trig functions also have inverses. The inverses of the sine, cosine, and tangent functions are written as , , and , respectively. [Be careful not to confuse the notation for the inverse sine function with .] These inverse functions are also sometimes written , , and , short for arcsine, arccosine, and arctangent, respectively. Your calculator should have three buttons with one of those sets of three labels. Since a trig function takes an angle and gives a ratio of sides, the inverse trig functions must take as input a ratio of sides and then give back an angle. For example, if you know that the length of the side adjacent to a particular angle is 12 and the length of the hypotenuse of this triangle is 13 , you can find the measure of angle using the inverse cosine. The cosine of would be 12/13, so the inverse cosine of 12/13 will give the value of : implies that . Using the or button on your calculator, you should check that the measure of is 22.6 . Part D The 3-4-5 right triangle is a commonly used right triangle. Use the inverse sine function to determine the measure of the angle opposite the side of length 3. Express your angle in degrees to three significant figures. 22.9 12/7/12 HW00: Introduction to MasteringPhysics 25/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« Hint 1. Using the inverse sine To use the inverse sine, first write down the formula for the sine of the angle: for the triangle in the figure. This tells you that the measure of the angle is the inverse sine of 3/5. ANSWER: Correct Part E A support wire is attached to a recently transplanted tree to be sure that it stays vertical. The wire is attached to the tree at a point 1.50 from the ground, and the wire is 2.00 long. What is the angle between the tree and the support wire? Express your answer in degrees to three significant figures. Hint 1. Choose the correct function Using the given information, which of the following functions should you use to find the measure of ? ANSWER: = 36.9 12/7/12 HW00: Introduction to MasteringPhysics 26/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« ANSWER: Correct Solving Quadratic Equations Learning Goal: To use the quadratic formula to solve quadratic equations. Maria wants to plant a small tomato garden in her yard. She bought 25 tomato plants, and she has read that ideally tomatoes are planted in a square grid to help them pollinate each other. Part A How many plants should she plant in each row so that her 25 plants end up in a square (i.e., plants in each of rows)? Express your answer as an integer. Hint 1. How to approach the problem If the gardener has plants in each of her rows, then the total number of plants will be . Since there are 25 total plants, the correct value of will satisfy the equation . Solve this equation by taking the square root of both sides. Note that 25 has both a positive and a negative square root. Since you can't have a negative number of plants, you only want the positive square root. ANSWER: = 41.4 5 12/7/12 HW00: Introduction to MasteringPhysics 27/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« Correct You may have been able to solve this part simply by intuition or with simple arithmetic. In the next part, you will use the quadratic formula to find the value of . Although this involves more work than necessary to solve this part, using the quadratic formula on a problem that you've already solved should help you to feel comfortable with it. There are several techniques for solving an equation of the form , where , , and are numeric constants, such as factoring and completing the square. The quickest way to solve such an equation that will work all the time is to use the quadratic formula. For an equation with the above form, the quadratic formula gives where the symbol means that there are two solutions: one obtained by replacing with a and the other obtained by replacing with a . The steps for using the quadratic formula are as follows: 1. Get your equation into the form . 2. Identify the values of , , and . 3. Substitute these values into the quadratic formula. 4. Simplify the resulting values for . From the previous part, if the gardener has plants in each of her rows, then the total number of plants will be . To make a square using 25 total plants, the value of must satisfy the equation . Part B To find in Part A, you would need to solve the equation . Which of the following shows the proper values in the quadratic formula before simplifying the radical and dividing? Hint 1. The values of , , and . is the value of the coefficient of , so in this problem, (because ). is the value of the coefficient of . Notice that the equation given has no term. Since , the coefficient of must be zero. Therefore, . Finally, is the constant term, so in this problem, . Plug these values into the quadratic formula and then reduce it to the form shown in one of the answer choices. ANSWER: Correct Notice that the value of is , not . You should always include the sign in the values of , , and . Also, notice that the answer implies there are two solutions to Part A: and . Of course, you can't have tomato plants, so for solving the practical problem of planting tomatoes, the only correct solution is 5 plants per row. 12/7/12 HW00: Introduction to MasteringPhysics 28/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« Part C Consider the equation . Plug the values for , , and into the quadratic formula, but do not simplify at all. Which of the following shows the proper substitution? ANSWER: Correct Part D Use the result from Part C to find the two solutions to the equation . Enter the two solutions separated by a comma. (The order is not important.) ANSWER: Correct Proportional Reasoning Learning Goal: To understand proportional reasoning for solving and checking problems. Proportional reasoning involves the ability to understand and compare ratios and to produce equivalent ratios. It is is a very powerful tool in physics and can be used for solving many problems. It's also an excellent way to check answers to most problems you'll encounter. Proportional reasoning is something you may already do instinctively without realizing it. Part A You are asked to bake muffins for a breakfast meeting. Just as you are about to start making them, you get a call saying that the number of people coming to the meeting has doubled! Your original recipe called for three eggs. How many eggs do you need to make twice as many muffins? Express your answer as an integer. ANSWER: Correct 2.50 , -1 6 12/7/12 HW00: Introduction to MasteringPhysics 29/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« Linear relationships Although this was a particularly simple example, you used proportional reasoning to solve this problem. It makes sense that if you need twice as many muffins, then you'd need twice as many eggs to make them. We say that the number of eggs is linearly proportional to the number of muffins. This sort of relationship is characterized by an equation of the form , where and are the two quantities being related (eggs and muffins here) and is some constant. In a situation where the constant is not important, we may write , which means " is proportional to ". Writing means we know that if the number of eggs triples, then the number of muffins triples as well. Or, if the number of muffins is divided by 5, then the number of eggs is divided by 5. The figure shows a graph of for some constant . You can see that when you double or triple the original value, you get double or triple the value, respectively. Keep this graph in mind and relate it to your intuitive sense as you solve the next problem. Part B You have a dozen eggs at home, and you know that with them you can make 100 muffins. If you find that half of the eggs have gone bad and can't be used, how many muffins can you make? Express your answer as an integer. ANSWER: Correct Recall that dividing a variable is the same as multiplying it by a fraction. If you keep this in mind, then you can change this problem from "the number of eggs are divided by two" into "the number of eggs are multiplied by one-half," which works just as any other multiplication. If you look at the graph for the linear relationship, dividing by 2 is like moving from the middle point to the left point marked on the graph. Quadratic relationships Quadratic relationships are also important in physics and many other areas. In a quadratic relationship, if one number is increased by a factor of , then the other is increased by a factor of . An example would be the relationship between area and radius of a circle. You know from geometry that . Since is a constant, you can rewrite this equation as , which says that is proportional to the square of . The relation applies to any equation of the form . The figure shows a graph of for some constant . You can see that when you double or triple the original value, you get four or nine times the value, respectively. 50 12/7/12 HW00: Introduction to MasteringPhysics session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« Part C When sizes of pizzas are quoted in inches, the number quoted is the diameter of the pizza. A restaurant advertises an 8- "personal pizza." If this 8- pizza is the right size for one person, how many people can be fed by a large 16- pizza? Express your answer numerically. Hint 1. How to approach the problem The area of a pizza is what determines how many people can be fed by the pizza. You know that the area of a circle is proportional to the square of the radius. Since the radius is proportional to the diameter, it follows that the area is also proportional to the square of the diameter: . Use this relation to determine how the area, and therefore the number of people fed, changes. ANSWER: Correct The stopping distance is how far you move down the road in a car from the time you apply the brakes until the car stops. Stopping distance is proportional to the square of the initial speed at which you are driving: . Part D If a car is speeding down a road at 40 ( ), how long is the stopping distance compared to the stopping distance if the driver were going at the posted speed limit of 25 ? Express your answer as a multiple of the stopping distance at 25 . Note that is already written for you, so just enter the number. Hint 1. Setting up the ratio Since , the car is moving at a speed 1.6 times the speed limit of 25 . The stopping distance is proportional to the square of the initial speed, so the stopping distance will increase by a factor of the square of 1.6. ANSWER: Correct The quadratic relationship between stopping distance and initial speed is part of the reason that speeding fines are doubled in school zones: At low speeds, a small change in speed can lead to a large change in how far your car travels before it stops. Inverse relationships A third important type of proportional relationship is the inverse relationship. In an inverse relationship, as one variable increases the other decreases and vice versa. For instance, if you had a $10 gift certificate to a chocolate shop, then the amount of chocolate that you could get would be inversely proportional to the price of the chocolate you picked. If you buy the $0.25 candies, you could get 40 of them, but if you opt to purchase candies whose price is higher by a factor of 4 ($1.00), then you must reduce the number that you get by a factor of 4 (to 10). Similarly, if the price decreases by a factor of 5 (to $0.05), then you increase the number by a factor of 5 (to 200). An inverse relationship is based on an equation of the form , where is a constant. If is inversely proportional to then you would write or . 4 = 2.56 12/7/12 HW00: Introduction to MasteringPhysics 31/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« or . The figure shows a graph of for some constant . You can see that when you double or triple the original value, you get one-half or one-third times the value, respectively. Part E A construction team gives an estimate of three months to repave a large stretch of a very busy road. The government responds that it's too much inconvenience to have this busy road obstructed for three months, so the job must be completed in one month. How does this deadline change the number of workers needed? Hint 1. The proportionality The time to complete the job should be inversely proportional to the number of workers on the job. Therefore, reducing the time by a factor of 3 requires increasing the number of workers by a factor of 3. ANSWER: Correct Inverse-square relationships All of these proportionalities are in some way familiar to you in your everyday life. There is one other important type in physics with which you may not be as familiar: the inverse-square relationship. The inverse-square relationship is based on an equation of the form , where is a constant. You would write or , either of which means " is inversely proportional to the square of ." Although this may look or sound more intimidating than the relations we've looked at previously, it works in essentially the same way. If is doubled, then is multiplied by one-fourth, and if is decreased by a factor of 2, is multiplied by 4. The figure shows a graph of for some . You can see that when increases by a factor of 2 or 3, decreases by a factor of 4 or 9, respectively. One-ninth as many workers are needed. One-third as many workers are needed. The same number of workers are needed. Three times as many workers are needed. Nine times as many workers are needed. 12/7/12 HW00: Introduction to MasteringPhysics 32/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« Part F The loudness of a sound is inversely proportional to the square of your distance from the source of the sound. If your friend is right next to the speakers at a loud concert and you are four times as far away from the speakers, how does the loudness of the music at your position compare to the loudness at your friend's position? ANSWER: Correct Inverse-square relations show up in the loudness of sounds, the brightness of lights, and the strength of forces. Proportional reasoning is useful for checking your answers to problems. If your answer is a formula, then you can explicitly check that all of the variables have the correct proportionalities. If you have a numerical answer, you can check your technique by doubling one of the starting variables and working through the same process to a solution. If your answer does not change as you expect it to based on the proportionality of the initial and final variables, then you know that something is wrong. Name That Line Learning Goal: To understand how to find the equation for a line using the slope-point and two-points techniques. Many equations in physics express linear relationships between two quantities, meaning that a graph of the two quantities would be a line. If you walk at constant speed, your position is linearly related to time. The graph of your position at various times might look like the graph in the figure. If you see that points on a plot have a linear relationship, you can then use a straight edge to draw in the line. From your graph, you can predict what other points lie on the line. However, usually the equation for the line will give you more accurate predictions. Then, you can find the values for other points on the line by putting numbers into the equation. Definition of slope The sound is one-sixteenth as loud at your position. The sound is one-fourth as loud at your position. The sound is equally loud at your position. The sound is four times as loud at your position. The sound is sixteen times as loud at your position. 12/7/12 HW00: Introduction to MasteringPhysics 33/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« All of the techniques for finding a line's equation use the definition of slope, which is given the symbol . Slope is defined as the difference between the y coordinates of two points, divided by the difference between the x coordinates of those two points. That is, if you have two points and on a line, the slope is . You might be able to remember the definition more easily in the form "the difference in the y values over the difference in the x values." The end result that you want is an equation that looks like , where is the slope and is the y intercept—the value of where the line intersects the y axis. For instance, you might have as the final result the equation . In the example given of distance walked, you could easily determine how far you had walked after 2.5 if you had an equation for the graphed line. Part A Suppose that you want to construct a line with slope that passes through the point . You would begin by setting up the equation . If you plug in the coordinates for any point on that line, the two sides of the equation will be equal. Once you've done this, you can solve for . What is the value of ? Express your answer as an integer. Hint 1. How to plug in the coordinates You have the equation and the point . You should put in the x coordinate where the is in the equation. This would give you . Then, put in the y coordinate where the is in the equation. This gives you . Now you have an equation with only one unknown: . Solve this for . ANSWER: = -5 Correct Putting this value back into the equation, you can see that the line you were looking for is . If you had just sketched a graph, like the one in the figure below, you likely could have seen that the point lies on this line, but you couldn't have checked a point such as with much accuracy. Another possible situation is that you would be given two points that lie on a line and would need to construct the equation for that line. To do so, you would use the same technique as in the previous part. The only difference is that you have to calculate the slope first. You would do this by using the same definition of slope that is given in this problem. Part B Suppose that you want to find the equation for a line that passes through the two points and . What is the slope of this line? Express your answer numerically. Hint 1. Find the change in y coordinates What is the difference in the y coordinates between the two points? Recall that the y coordinate is the second number in the ordered pair. Express your answer numerically. ANSWER: Hint 2. Find the change in x coordinates What is the difference in the x coordinates between the two points? Recall that the x coordinate is the first number in the ordered pair. Express your answer numerically. ANSWER: ANSWER: Correct 6 4 = 1.50 12/7/12 HW00: Introduction to MasteringPhysics 35/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« Part C Now that you have the slope, proceed as you did in Part A. What is the value of for this line? Express your answer numerically. Hint 1. Which point to plug in Since the equation for a line is true for any point on the line, you can plug either point in. Using will make things a bit easier, but plugging in either point should allow you to easily solve for . ANSWER: Correct Combining the answers from Parts B and C gives an equation for the line of . The graph of this line would look like the one in this figure. Part D Now, find the equation for a line that passes through the two points and . Express your answer in terms of . The " " has been given for you. Hint 1. Find What is the slope of the line? Recall that slope is the difference in the y values over the difference in the x values. Express your answer numerically. ANSWER: Hint 2. Find Now, you have = 3 = -1.60 12/7/12 HW00: Introduction to MasteringPhysics session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« . Substitute in the coordinates from one point so that you can solve for the value of . What is the value of ? Express your answer numerically. ANSWER: ANSWER: Correct The graph of this line would look like the one in this figure. . Part E What is the y coordinate of the point on the line with x coordinate 2? Express your answer numerically Hint 1. How to approach the problem Since the equation is true for all points on the line, if you plug in the x coordinate for in the equation, then you can solve for , and the number that you get will be the y coordinate of that point. This procedure is made easier by the fact that the equation is already solved for ! ANSWER: Answer Requested Multiplying and Dividing Fractions = 1.40 = = 4.60 12/7/12 HW00: Introduction to MasteringPhysics 37/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« Learning Goal: To understand the multiplication and division of fractions. If you had eight quarters, you could likely figure out relatively quickly that this amounted to two dollars. Although this may be purely intuitive, the underlying math involves multiplication of fractions. The value of eight quarters is the same as . When you multiply fractions, you multiply the numerators (the top numbers in the fractions) to get the numerator of the answer, and then multiply the denominators (the bottom numbers in the fractions) to find the denominator of the answer. In this example, , giving 2, as you expected. Similarly, asking for a fraction of a fraction (e.g., "one fifth of a quarter") is a case of multiplying fractions: . In this problem, before entering your answer, be sure to reduce your fraction completely. If you get 8/ 6 for your answer, reduce it to 4/3 before entering it, or else it will be marked wrong. Also, don't worry if the numerator is larger than the denominator. It is almost always easier and more useful to further calculations to leave such answers as improper fractions rather than to convert them to mixed numbers such as . Part A If you have a quarter of a pie and you cut it in half, what fraction of a pie would each slice represent? Give the numerator fol lowed by the denominator, separated by a comma. Hint 1. Setting up the equation The problem is asking you for the value of "half of a quarter." This translates into math as . Multiply the numerators to find the numerator of the answer. Then, multiply the denominators to find the denominator of the answer. ANSWER: Correct Part B Find the value of . Though these numbers aren't quite as nice as the ones from the example or the previous part, the procedure is the same, so this is really no more difficult. Give the numerator fol lowed by the denominator, separated by a comma. Hint 1. Find the numerator To find the numerator, simply multiply the numerators of the two fractions (17 and 2). What is the result? ANSWER: 1 , 8 34 12/7/12 HW00: Introduction to MasteringPhysics 38/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« Hint 2. Find the denominator To find the denominator, simply multiply the denominators of the two fractions (15 and 11). What is the result? ANSWER: ANSWER: Correct Dividing fractions is no more difficult than multiplying them. Consider the problem . Notice that division by 3 is identical to multiplication by 1/3, because both operations consist of breaking the first number into three parts. Thus . The only new step in division of fractions is that you must invert (flip) the second fraction. Then, simply multiply as shown here. Part C Consider the following equivalent expressions: and . What are the values of and ? Give the value of fol lowed by the value of , separated by a comma. ANSWER: Correct Part D Calculate the value of . Give the numerator fol lowed by the denominator, separated by a comma. ANSWER: Correct Part E Now, find the value of 165 34 , 165 12 , 13 9 , 26 12/7/12 HW00: Introduction to MasteringPhysics session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« . Don't be intimidated by the complexity of this expression. Finding this value consists of simply multiplying twice and then dividing once, tasks that are no more difficult than what you've done before. Give the numerator fol lowed by the denominator, separated by a comma. Hint 1. How to approach the problem Break the problem down into individual operations. Notice that is the same as . This second form makes the individual operations more obvious. First, multiply . Then, multiply . Finally, divide the two fractions that you have found. Hint 2. Find the value of What is the value of ? Give the numerator fol lowed by the denominator, separated by a comma. ANSWER: Hint 3. Find the value of What is the value of ? Give the numerator fol lowed by the denominator, separated by a comma. ANSWER: ANSWER: Correct Interpreting Graphs 3 , 40 21 , 8 1 , 35 12/7/12 HW00: Introduction to MasteringPhysics 40/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« Learning Goal: To be able to gain many different types of information from a graph. Being able to read graphs is an important skill in physics. It is also critical in day-to-day life, as information in the news and in business meetings is often presented in graphical form. In this problem, you will consider a single graph and all of the information that can be gained from it. Since the graph axes have no labels, think of it as a graph of something important to you, whether that is GPA, your bank balance, or something else. Specific applications will be noted for each way of analyzing a graph. The easiest information to obtain from a graph is its value at a point. The height of the graph above the horizontal axis gives the value of the graph. Points above the horizontal axis have positive values, whereas points below the axis have negative values. The vertical axis will usually have specific values marked off so that you can tell exactly what value each height corresponds to. In the graph you've been given, there are no exact values labeled, but you can still tell relative values; you can make statements such as, "At point D, the graph has a greater value than at point C." Part A At which point(s) does the graph have a positive value? Enter al l of the correct letters in alphabetical order. For instance, i f you think that the correct choices are B and F, you would enter %). Hint 1. Determining positive values from a graph Any value above the horizontal axis is positive; any below is negative. Where the graph intersects the horizontal axis, the value of the graph is zero, which is neither positive nor negative. ANSWER: Correct The difference between positive and negative is important in many situations, for instance on your bank statement. In physics it makes a big difference in many scenarios. Positive position means to the right or above some reference point; negative position means to the left or below the reference point. Positive velocity means moving to the right, whereas negative velocity means moving to the left. The graph is often more convenient than a table of numbers or an equation, because you can immediately see where the graph takes positive values and where it takes negative values. With an equation or a table of numbers, this would take some algebra or guess work. Since the value of the graph at a point is indicated by its height above the horizontal axis, the maximum value of the graph is the highest point on the graph. Similarly, the minimum value of the graph is the lowest point, which may be below the horizontal axis. Part B At which point does the graph have its maximum value? Enter the correct letter. ANSWER: ABCDEF E Correct Rate of change and slope Another type of information that can be gleaned from a graph is the rate of change of the values. Just as you care whether your bank account has a positive or negative value (i.e., if you have money or owe the bank money) you may also be interested in the rate of change of your bank account. If the rate of change is positive, then you are gaining money. If the rate of change is negative, then you are losing money. The rate of change of a graph is given by the slope of the graph. If the graph is a line, then the slope is just the slope that you are accustomed to for lines (i.e., the change in the vertical position divided by the change in the horizontal position). There are a few important things to remember about slope. If the line tilts upward as you follow it to the right (like this: ) then the graph has a positive slope. We would say that the graph is increasing (becoming more positive). Similarly, if it tilts downward as you follow it to the right (like this: ) then the slope is negative and we say that the graph is decreasing (becoming more negative). The steeper the tilt, the larger the rate of change." Part C Look at the graph from the introduction. The three points C, D, and F are all on straight segments. Rank them from greatest rate of change to least rate of change. Hint 1. Slope at point D When the graph is horizontal, the slope of the graph is zero. You can see this by noticing that the change in vertical position as you move from left to right across a horizontal graph is zero. No matter what the change in horizontal position is, when you divide to find the slope it will be zero. ANSWER: Correct For more complex curves, you will have to draw the tangent line at a point to determine the rate of change of the graph at that point. The tangent is a line that just touches the curve. To do this, instead of passing through two nearby points, the line has to align itself so that its rate of change is the same as the rate of change of the graph at that point. Therefore, once you've drawn the tangent line, its slope is the same as the slope of the curve at 12/7/12 HW00: Introduction to MasteringPhysics 42/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« that point. In general, you will be able to rely on your intuitive sense of "Is the graph growing higher or lower at this point?" but it's good to keep in mind this more precise definition in terms of the tangent line. It will help you out in situations that are hard to figure out by simple examination. In the following two parts, consider again the graph shown in the introduction to the problem. Part D At which point is the graph increasing at the greatest rate? For now, ignore point E. We will discuss it after this part. Enter the correct letter. Hint 1. Drawing the tangent To find the tangent line at a particular point, you should draw a dot at that point on the graph and then draw pairs of points, one on either side of the point you care about, that are the same distance from the point you care about. If you then connect those pairs of dots, the lines connecting them will get closer and closer to the proper slope as you move to pairs that are closer and closer to the point you care about. Once you get pretty close to that point, you should be pretty confident of the slope for the tangent line. With practice, you will gain an intuitive ability to see roughly how the slope of the tangent at a point should look. ANSWER: D 12/7/12 HW00: Introduction to MasteringPhysics 43/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« Correct You were told to ignore point E for this part. This is because the rate of change is not well defined at sharp corners. You won't ever be asked for the rate of change of a graph at a sharp corner, though points near the corner should have well-defined rates of change. Points B and C are also special, because the slope at those points is zero. This should be easy to see at C, since the graph is actually a horizontal line in the area near C. If you carefully work out the tangent at point B using the method described in the hint for this part, you will see that the tangent is horizontal there as well. Since a horizontal tangent has a slope of zero, which is neither negative nor positive, the graph is neither increasing nor decreasing at points B and C. Part E At which point(s) is the graph decreasing? Enter al l of the correct letters in alphabetical order. For instance, i f you think that the correct choices are B and F, you would enter %). ANSWER: Correct Area under a graph The other piece of information important to physics that can be found from a graph is the area under the graph between two points on the graph. The light blue region in the figure shows the area under the graph between two points. The area under a graph is important if you have graphed the rate of change in some quantity. In physics, you might have a graph of the velocity of some object vs. time. Since velocity is the rate of change of position, the area under the velocity graph between two times gives the total change in position between those two times. An important point is that if the graph dips beneath the horizontal axis, then the area below the axis is subtracted from the area above the axis. In this figure , the same graph is shown, but now the area is between two more widely separated points. However, if you compare this figure to the previous one, you'll see that equal areas were added above and below the horizontal axis. Therefore, the "area under the graph" in the two figures is the same, even though you see more shading in the second figure. In the following two parts, consider again the graph shown in the introduction to the problem. Part F You wish to find the area under the graph between the origin and some point on the graph. Which point will yield the greatest area? Enter the correct letter. ANSWER: Correct Part G FGH G 12/7/12 HW00: Introduction to MasteringPhysics 44/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« You are looking at the area under the graph between two points. The area is zero. Which two points are you looking at? Enter the two letters in alphabetical order. For instance, i f you think that the correct choice is B and F, you would enter %). Hint 1. How to approach the problem Since you want the area under the curve between two points to be zero, the graph must define equal-sized shaded regions above and below the horizontal axis. Look for two points that are near each other, one above and one below the horizontal axis. ANSWER: Correct Pay Up! Learning Goal: To learn to solve linear equations. Almost every topic in physics will require you to solve linear equations—equations that don't contain any higher powers of the variable such as , , etc. Linear equations are the simplest algebraic equations. They arise in all sorts of situations. For this problem we'll look at one that might come up in your daily life. Suppose that you and three friends go out to eat and afterward decide to split the cost evenly. Your friend Anika points out that she only had a drink, so she should pay less ($2, the cost of her drink) and the rest of you can split the remainder of the bill. A linear equation can easily determine how much each of you must pay. For the particular problem raised in the introduction, assume that the total bill is $44. To answer the question "How should the bill be split?" we will create a linear equation. The unknown is how much money a single person (besides Anika) must pay, so call that . Although four people (you plus three friends) went to dinner, only three are paying the unknown amount for a total of . Since Anika is paying $2, the total amount paid is dollars, which must equal the amount of the bill, $44. Thus, the equation to find is . The steps for solving a linear equation are as follows: 1. Move all of the constants to the right side. 2. Move all of the variable terms (terms containing ) to the left side. 3. Divide both sides by the coefficient of the variable to isolate the variable. You will go through these steps one at a time to solve the equation and determine how much each person should pay. Part A The first step in solving a linear equation is moving all of the constants (i.e., numbers like 2 and 44 that aren't attached to an ) to the right side. What is the final value on the right side once you've moved all of the constants? Express your answer as an integer. Hint 1. How to "move" a constant The term move is not exactly an accurate description of how you get rid of the constant on one side of an equation. What you actually do is add the opposite of that number to both sides. In this way, the constant no longer appears on the side that you don't want it on. For instance, if you have , you want to move the 3 to the right. The opposite of 3 is 3, so you would subtract 3 from both sides: , which reduces to . FH 12/7/12 HW00: Introduction to MasteringPhysics 45/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« Similarly, if you had , you would add 12 to each side, because the opposite of 12 is 12, and so , . ANSWER: Correct Part B Now that you have , you need to isolate the variable so that you have an equation of the form " some number." What is the value of (i.e., the amount you must pay)? Express your answer as an integer number of dollars. Hint 1. How to isolate the variable To remove the coefficient from the variable, simply divide both sides of the equation by that coefficient. For instance, if you had , then you would divide both sides by 2: , yielding . ANSWER: Correct The next problem looks more intimidating, but it requires the same procedures: Move all constants to the right, then move all variables to the left, and finally divide both sides by the variable’s coefficient. Part C If , what is the value of ? Express your answer as an integer. Hint 1. Collect the constant terms You have the equation and need to get all of the constants on the right side. Which of the following would get all of the constant terms on the right side of the equation? ANSWER: = 42 = 14 12/7/12 HW00: Introduction to MasteringPhysics 46/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« Hint 2. Collect the terms with After adding 23 to both sides, you have the equation and need to get all of the terms with on the left side. Which of the following would get all of the variable terms on the left side? ANSWER: ANSWER: Correct Part D If , what is the value of ? Express your answer as an integer. Hint 1. Collect the constant terms You have the equation and need to get all of the constants on the right side. Which of the following would get all of the constant terms on the right side of the equation? ANSWER: Hint 2. Collect the terms with After subtracting 9 from both sides, you have the equation .You need to get all of the terms with on the left side. Which of the following would get all of the variable terms on the left side? ANSWER: Add 23 to both sides. Subtract 23 from both sides. Add 22 to both sides. Subtract 22 from both sides. Add to both sides. Subtract from both sides. Add to both sides. Subtract from both sides. = 5 Add 9 to both sides. Subtract 9 from both sides. Add 19 to both sides. Subtract 19 from both sides. 12/7/12 HW00: Introduction to MasteringPhysics 47/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« ANSWER: Correct Converting Units The ability to convert from one system of units to another is important in physics. It is often impractical to measure quantities in the standard meters, kilograms, and seconds, but the laws of physics that you learn will involve constants that are defined in these units. Therefore, you may often have to convert your measured quantities into meters, kilograms, and seconds. The following table lists metric prefixes that come up frequently in physics. Learning these prefixes will help you in the various exercises. mega- ( ) kilo- ( ) centi- ( ) milli- ( ) micro- ( ) nano- ( ) When doing unit conversions, you need a relation between the two units. For instance, in converting from millimeters to meters, you need to know that . Once you know this, you need to divide one side by the other to obtain a ratio of to : . If you are converting from millimeters to meters, then this is the proper ratio. It has in the denominator, so that it will cancel the units of the quantity that you are converting. For instance, if you were converting , then you would have . If you were converting a quantity from meters to millimeters, you would use the reciprocal ratio: . Part A Suppose that you measure a pen to be 10.5 long. Convert this to meters. Express your answer in meters. Add to both sides. Subtract from both sides. Add to both sides. Subtract from both sides. = -4 12/7/12 HW00: Introduction to MasteringPhysics 48/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« Hint 1. Relating centimeters and meters To solve this problem, you will need to use the relation . You can determine such relations using the metric prefixes given in the introduction to this problem. If one centimeter equals meters, then you need centimeters to equal a whole meter, just as you know that if one quarter equals US dollars, then you need quarters to equal a whole US dollar. ANSWER: Correct When converting areas, you must be careful to use the correct ratio. If you were converting from to , it might be tempting to use again. Be careful! Think of as . That is to say, think of this as a pair of millimeter units, each of which must be converted separately. To convert to square meters you would use the following calculation: . Notice that the exponent distributes to both the numbers and the units: . Now the will cancel properly: . Part B Suppose that, from measurements in a microscope, you determine that a certain bacterium covers an area of . Convert this to square meters. Express your answer in square meters. Hint 1. Find the conversion factor Which of the following gives the proper conversion factor to use? From the table in the introduction, you can see that , which gives . ANSWER: 10.5 = 0.105 12/7/12 HW00: Introduction to MasteringPhysics 49/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« ANSWER: Correct As with areas, you must be careful when converting between volumes. For volumes, you must cancel off three copies of whatever unit you are converting from. Part C Suppose that you find the volume of all the oceans to be in a reference book. To find the mass, you can use the density of water, also found in this reference book, but first you must convert the volume to cubic meters. What is this volume in cubic meters? Express your answer in cubic meters. Hint 1. Find the conversion factor Which of the following gives the proper conversion factor to use? From the table in the introduction, you can see that . ANSWER: ANSWER: 1.50 = 1.50×10 −12 12/7/12 HW00: Introduction to MasteringPhysics 50/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« Correct Part D In a laboratory, you determine that the density of a certain solid is . Convert this density into kilograms per cubic meter. Notice that the units you are trying to eliminate are now in the denominator. The same principle from the previous parts applies: Pick the conversion factor so that the units cancel. The only change is that now the units you wish to cancel must appear in the numerator of the conversion factor. Express your answer in ki lograms per cubic meter. Hint 1. Find the conversion factor Which of the following gives the proper conversion factor to use? From the table in the introduction, you can see that . Recall that you are trying to cancel units out of the denominator of the fraction. ANSWER: ANSWER: Correct You are now ready to do any sort of unit conversion. You may encounter problems that look far more complex than those you've done in this problem, but if you carefully set up conversion factors one at a time to cancel the units you don't want and replace them with the units that you do want, then you will have no trouble. Derivatives Learning Goal: To understand the concept of derivatives as the slope of a function graph. There are two fundamental tools of calculus—the derivative and the integral. The derivative is a measure of the rate of change of a function. You’ll see derivatives often throughout your study of physics because much of physics involves describing rates of change. = 1.40×10 18 = 5230 derivatives often throughout your study of physics because much of physics involves describing rates of change. Part A What does “rate of change” mean, exactly? A function describes how one quantity is related to another – how y is related to x. As an example, y could be your height and x your age. The three graphs below show three cases of how your height could change with age. Sort the graphs according to which best represents the graph of your height against age for the following periods: During your childhood, as you grew up Between age 30 and age 50 During old age, when your spine compacts a little Drag the appropriate items to their respective bins. Hint 1. Identify the graph that illustrates growth Age is plotted on the horizontal axis in each graph, with age increasing from left to right. Height is plotted on the vertical axis. Think about the childhood period, when you are growing. What kind of graph illustrates growth? ANSWER: ANSWER: A graph that rises as you move from left to right. A graph that is flat as you move from left to right. A graph that falls as you move from left to right. Correct The positive (upward) slope on Graph 2 corresponds to your increasing height during childhood. Your height won’t change between ages 30 and 50, so Graph 1 is correct: It is horizontal, and its slope is zero. And finally, your decreasing height during old age corresponds to the negative (downward) slope on Graph 3. These three graphs of height vs. age are three important special cases of functions: Graph 1 shows a quantity that remains constant with respect to another quantity. Graph 2 shows a quantity that increases with respect to another quantity. Graph 3 shows a quantity that decreases with respect to another quantity. These graphs also give three important special cases in our understanding of rate of change. Part B Graphs 4 and 5 show the same information as Graphs 2 and 3, but this time with specific information about height and age on the graph axes. From the graphs, you can find the exact rate of change of the person’s height in each case. Find the rate of change of the height from age 8 to 16 and from age 60 to 80. Express both answers in units of cm per year, to two significant figures. Separate your answers with a comma. Express your answers numerically to two significant figures separated by a comma. Hint 1. How to approach the problem The rate of change corresponds to the slope on the graph. The slope is commonly thought of as rise/run. In terms of the x and y values on the graph, the slope is Read the values of and from the graph, and from these compute the slope as given by the formula above. In this problem, is the change in age and is the change in height. Hint 2. Calculate the slope of Graph 4 It’s often easiest to pick points on the line that lie on the intersection of gridlines: this makes reading off the values on the axes straightforward. On Graph 4, we have two such points: and . What are the rise and run values corresponding to these two points? Express your answers numerically separated by a comma. ANSWER: Hint 3. Calculate the slope of Graph 5 It’s often easiest to pick points on the line that lie on the intersection of gridlines: This makes it straightforward to read off the values on the axes. On Graph 5, we have two such points: and . What are the rise and run values corresponding to these two points? Express your answers numerically separated by a comma. ANSWER: rise, run = 50 , 8 ANSWER: Correct What you’ve just done is calculate the derivative of a function, in this case a person’s height vs. time, because the derivative measures a function’s rate of change. The answers here illustrate that a derivative can have a positive value, when the function is increasing; a negative value, when the function is decreasing; a value of zero, when the function isn’t changing. These are the three special cases of rates of change we mentioned above. Part C The next graph shows a different function. Which of the graphs below is the derivative of this function? Hint 1. How to approach the problem Notice that the slope of the graph in Part C is constant. So you’re looking for an answer graph that’s constant - that is, a graph with a horizontal slope. Hint 2. Narrowing down the options Which of the two graphs with horizontal slopes is correct? To determine this, measure carefully the slope of your graph, using the numbers provided on the axes. Then, determine the slope as you did in Part B. ANSWER: rise, run = -5 , 20 6.3 , -0.25 HW00: Introduction to MasteringPhysics 54/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« Correct The original function is increasing at a constant rate throughout the interval, so the answer must be a constant. Measuring the slope of the graph carefully leads you to the correct answer, 0.6. Part D So far we have only looked at functions with a constant slope (positive, negative, or zero). But the slope of a function, and thus its derivative, can change. Consider the next function shown. Rank, in increasing order, the derivatives of the function at each of the points marked A through D. Express your answer in order of increasing slopes as a string w ithout spaces between points. For example, enter $%&D i f you think the derivative at point A is the smallest, B is the next larger, and so on. Hint 1. How to approach the problem Look carefully at the graph’s slope at each of the four points. The slope may be positive, zero, or negative. If you are ranking in the order of increasing slopes, then negative slopes will be listed first, then zero, and then positive slopes. For a positive slope, the more sharply the graph slopes upward, the larger the slope. 12/7/12 55/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« graph slopes upward, the larger the slope. ANSWER: Correct The slope at point C is negative, so the derivative there is the smallest. At B, the graph is horizontal, so the derivative is zero. At points A and D, the steeper slope corresponds to the larger derivative. Part E Functions are not all straight lines. If a function is curved, the derivative is not constant but changing. To find the derivative at any point, draw a line tangent to the graph at that point. The slope of the tangent line is the derivative. As you move along the graph, the slope of the tangent line changes, and so does the derivative. The graph below shows a function ( ). The derivative of the function at is closest to: Hint 1. Narrowing down the choices The derivate at is the slope of the curve there. Is the function increasing, decreasing, or zero at ? ANSWER: Hint 2. Finding the answer To determine the exact derivative of the function at , draw a line tangent to the graph at that point. The slope of the tangent line tells you the function’s derivative. ANSWER: CBDA decreasing increasing zero –5 5 10 0 –10 12/7/12 HW00: Introduction to MasteringPhysics 56/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« Correct At , the function is increasing, so that tells you that the derivative (the slope of the tangent line) is positive. Here, you needed to measure the slope of the tangent line carefully to find the numerical answer. Part F Referring again to the graph in Part E, rank, in increasing order, the derivatives of the function at each of the points A through E. If two of the values are equal, you may list them in either order. Express your answer in order of increasing slopes as a string w ithout spaces between points. For example, enter $%&DE i f you think the derivative at point A is the smallest, B is the next larger, and so on. Hint 1. How to approach the problem Assess the values of the derivatives by looking carefully at the graph’s slope at each of the five points. The slope may be positive, zero, or negative. To rank the derivatives in increasing order, the negative derivatives (negative slopes) will be listed first, then zero, and then the positive derivatives (positive slopes). For a positive slope, the more sharply the graph slopes upward, the larger the slope and hence the larger the derivative. ANSWER: Correct At points B and D, the tangent line is horizontal, so the derivative is zero. At point C, the tangent line slopes downward, and the derivative is negative. Thus, point C has the smallest derivative, followed by B and D, which are equal. Measuring the two positive derivatives (the slopes of the tangent lines at A and E) reveals that the slope (derivative) at E is smaller than at A. In physics, ( ) might represent an object’s position in one dimension versus time. In that case, the derivative represents velocity. Positive and negative velocity correspond to motion in opposite directions, and zero velocity means the object is at rest. This is just one of the important ways that derivatives are used in physics. Integrals Learning Goal: To understand the concept of an integral as the area under a function graph. There are two fundamental tools of calculus—the derivative and the integral. Just as the derivative can be understood in a visual way as the slope of a function graph, what is called a definite integral can be understood as the area under a function graph. Definite integrals have many applications in physics, such as finding an object’s displacement from a known velocity and finding work done by a known force. A definite integral gives you a numerical answer. There is a second kind of integral, called the indefinite integral (or antiderivative), which is itself another function, but we won’t discuss that kind in this tutorial. Part A Many practical applications require calculations of area. Some of these calculations are straightforward, but others are more difficult, and it turns out that the idea of the definite integral can help us with the more difficult cases. Let’s start with some familiar shapes. What is the area of a rectangle of length and width ? Express your answer in terms of and . ANSWER: CBDEA 12/7/12 57/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« Correct The area of a rectangle is found by simply multiplying the length by the width. You also know how to find the area of a circle. But if you didn’t already know the formula, how would you go about figuring it out? In Part B, we’ll look at one method in detail to introduce key ideas about integrals. Part B Take a circle of radius as shown in, and imagine peeling it like an apple: Cut off a thin strip from all the way around the edge (the green strip), and straighten that strip out to a rectangle, as shown in . Call the tiny width of the strip . The length of the strip is because that’s the circumference of the circle. The bit of “peel” is the circumference of the skin on the outside of the circle. What is the area of the green strip? Express your answer in terms of , , and . Hint 1. How to approach the problem Notice that the strip in question becomes a rectangle in the figure Part B.2. From Part A, the area of a rectangle is its length times its width. ANSWER: Correct The area of this rectangle is its length multiplied by its width . Part C Now, imagine peeling more and more strips of width from our circle, and laying them all out as rectangles side by side, as shown in – starting with the first (outermost) strip at the far right. As you peel more and more, the strips get shorter and shorter because the circle is getting smaller and smaller, until you have the smallest strip laid at far left – this strip, if is really tiny, is essentially the center of the circle! What is the area of the circle? You’ve now chopped it into little strips, but notice that the area of the circle is the same as the sum of the areas of all our strips. If is really small, you can see that all the strips approximate a triangle in the figure Part C (the area under the diagonal line). So the sum of the areas of all our strips is the area of this triangle. What is the area of the triangle? Express your answer in terms of and . 12/7/12 HW00: Introduction to MasteringPhysics 58/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« Hint 1. The formula for the area of a triangle Keep in mind that the area of a triangle is one half its base times its height. ANSWER: Correct The area of the triangle is half the base (which is ) times the height (which is ). The area of the triangle in the figure Part C is , the formula you know for the area of a circle. We’ve proved it! We didn’t do it entirely rigorously, but the essence of our method (dividing something into tiny pieces and adding those up) lies at the heart of understanding definite integrals. Part D Now, look at a general function , as shown. Suppose you want to find the area under the curve, between and —that is, the blue shaded area. Using our method above, we can add up the areas of lots of skinny rectangles like the green one. Each rectangle has area , and we want to sum these areas between and , letting get very small. Mathematically, you may recognize this sum as a definite integral: . The notation may look intimidating, but just keep in mind that the refers to the small rectangle areas, the sign stands, in a sense, for the S in “sum” (that’s actually how the symbol for integration was chosen), and the and at the bottom and top of give you the start and end of the region you’re interested in. Thus, the definite integral means the total area under the curve between and . For the function shown below, find the definite integral . Express your answer numerically. Hint 1. How to approach the problem is the area under the graph between and . Because this function graph is horizontal between and , the area you need to find is that of a rectangle. 12/7/12 HW00: Introduction to MasteringPhysics 59/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« ANSWER: Correct The area under this function curve is a rectangle, so its area is just its length times its width. Part E For the function shown below, find the definite integral . Express your answer numerically. Hint 1. The areas below the x-axis Again, in this part you are finding the area of a rectangle, but when the function graph is below the x-axis, the area is counted as negative. ANSWER: Correct Just as in Part D, the area is that of a rectangle, so it’s straightforward to find. The twist here is that the area lies below the x-axis, so that area is counted as negative. Part F For the function shown below, find the definite integral . 15 -6 12/7/12 60/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« Express your answer numerically. Hint 1. How to approach the problem This time you have several rectangles. Find the area of each and add them, keeping in mind that area below the x-axis is negative. ANSWER: Correct In this case, the net area is the sum of positive area (above the x-axis) and negative area (below the x-axis). Part G Find the definite integral of the function shown in the graph below, over the limits from 0 to 3. Express your answer numerically. Hint 1. How to approach the problem In this case the function varies, but you can still find the definite integral by finding the area under the graph. That area is a triangle, so its area is half the base multiplied by the height. ANSWER: 8 18 12/7/12 HW00: Introduction to MasteringPhysics 61/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« Correct In this case, the function varies, but you can still find the definite integral by finding the area under the graph. That area is a triangle, so its area is ½(base)(height) = ½(3)(12) = 18. One application of definite integrals in physics is in calculating the work done by a force acting through a displacement. If the function shown in the figure represents force, then the definite integral is the work done by that force. In this example, if the force is measured in newtons ( ) and position is in meters ( ), then the work done, measured in joules ( ), is Part H For the function shown below, the definite integral is closest to which of the following? Hint 1. How to approach the problem This problem involves adding different kinds of area. Break the entire interval (from to ) into smaller intervals to find shapes that you recognize. Find the area of each, and then add your results to find the net area (the definite integral). Hint 2. What are the shapes? As shown below, this function can be broken into rectangles (yellow) and triangles (green). The area of each rectangle is length times width, and the area of each triangle is half its base times height. Read the lengths you need from the values on the graph. Hint 3. The positive and negative areas Recall that area above the x-axis is counted as positive, and area below the x-axis is negative, as shown below. 12/7/12 HW00: Introduction to MasteringPhysics 62/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« ANSWER: Correct In this case, you had to break the larger interval into smaller ones to identify the different shapes and find their areas. The definite integral always corresponds to the net area over the entire interval. Introduction to Significant Figures Many questions in Mastering require you to enter your answer to a specified number of significant figures. For grading purposes, some numeric answers must be exact. Suppose you are asked "How many days are in a week?" The answer must be "7." Other numeric answers may be graded as correct not only when they match the exact answer but also if they fall within an acceptable range of the exact answer. This range is known as the grading tolerance. Suppose you are asked "How many days are in a year?" The precise answer is 365.24 days. However, a calendar contains 365 days, so the Mastering system will also accept that as correct. An answer that falls within the Mastering grading tolerance will also be marked as correct and you will be given full credit. You will see a feedback box that looks like the one below: Because you rounded differently than the system did, your rounded answer might affect your work on further calculations in the same item and could cause your next answer to fall outside of the tolerance. To prevent this, you will be asked to use the unrounded answer for all future parts. Part A Suppose you are asked to find the area of a rectangle that is 2.1- wide by 5.6- long. Your calculator answer would be 11.76 . Now suppose you are asked to enter the answer to two significant figures. (Note that if you do not round your answer to two significant figures, your answer will fall outside of the grading tolerance and be graded as incorrect.) Enter your answer to two significant figures and include the appropriate units. ANSWER: –4 0 4 –8 8 12 12/7/12 HW00: Introduction to MasteringPhysics 63/64session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView assignmentID=« Correct This is your final answer, rounded to two significant figures. To calculate an answer to the correct number of significant figures, you must complete all calculations first and then round your final answer as the very last step. In Part C, you are asked to calculate the volume of a rectangular prism that has a length of 5.6 , a width of 2.1 , and a height of 6.6 . You can calculate the volume of a prism by multiplying the area of the base times the height. You already calculated the area of the base as in Part A. Part B What value should you use as the area of the base when calculating the answer to Part C? ANSWER: Correct 11.76 is the correct, unrounded value of the area of the base. It is correct to round to the requested number of significant figures in Part A because calculating the area is your final step. If you want to use that information in further calculations you must use the unrounded value. Part C Using the correct answer from Part B, calculate the volume of a rectangular prism with a length of 5.6 , a width of 2.1 , and a height of 6.6 . Enter your answer to two significant figures and include the appropriate units. ANSWER: Correct Part D Would an answer that generated the response below be considered correct and give you full credit? 12 11.76 11.8 78 12/7/12 HW00: Introduction to MasteringPhysics 64/64 ANSWER: Correct Score Summary: Your score on this assignment is 118%. You received 10.93 out of a possible total of 11 points, plus 2.05 points of extra credit. It is impossible to tell. No. Yes.