Chapter 2 Motion Along a Straight Line Goals for Chapter 2 To study motion along a straight line To define and differentiate average and instantaneous velocity To define and differentiate average and instantaneous acceleration To explore applications of straight-line motion with constant acceleration To examine freely falling bodies Introduction This study of motion … kinematics … is common to our lives yet full of interesting features. Displacement, time, and the average velocity—Figure 2.1 Displacement: a change in position of object Average x-velocity, (Page 37) Average velocity between t1 and t2? The track, its motion, and a graph—Figures 2.2 and 2.3 Motion may be analyzed graphically to understand the changes that are occurring and the data that can be extracted. Displacement & distance and velocity & speed Distance is a scalar quantity which refers to "how much path an object has covered" during its motion. Displacement is a vector quantity which refers to "how far out of place an object is"; it is the object's overall change in position. Velocity is a vector quantity which refers to "the rate at which an object changes its position." Speed is a scalar quantity which refers to "how fast an object is moving." , where s stands for distance. Displacement & distance and velocity & speed t=0s t=2s t=4s t=6s t=8s Average and instantaneous velocities—Figure 2.4 Instantaneous velocity is the velocity at any specific instant of time or specific point along the path. Instantaneous x-velocity: A safari and a chase—Figure 2.6 Refer to Example 2.1 and Figure 2.6. (Page 41) Cheetah’s coordinate x = 20 m + (5.0 m/s2) t2 (c) Instantaneous velocity at t1 = 1.0 s with t = 0.011 and 0.0001 s (d) A general expression for instantaneous velocity x-t curve and instantaneous x-velocity (a) (b) (c) Follow the motion of a particle—Figure 2.8 Motion diagram: the particle’s position at various times as well as arrows to represent the particle’s velocity at each instant. The motion of the particle may be described at selected moments in time. Q2.2 This is the x–t graph of the motion of a particle. Of the four points P, Q, R, and S, the speed is greatest at A. point P. B. point Q. C. point R. D. point S. E. not enough information in the graph to decide The average acceleration—Example 2.2 Acceleration is a vector quantity describing the rate of change of velocity with time. Average x-acceleration: (Page 44) The average x-acceleration between (a) t1 = 1.0 s to t2 = 3.0 s (c) t1 = 9.0 s to t2 = 11.0 s Speed increases or decreases ? Instantaneous x-acceleration: Finding the acceleration—Figure 2.12 A graph of and t may be used to find the acceleration. Find the slope of a tangent line at any given point. This is the vx–t graph for an object moving along the x-axis. Which of the following descriptions of the motion is most accurate? Q2.9 A. The object is slowing down at a decreasing rate. B. The object is slowing down at an increasing rate. C. The object is speeding up at a decreasing rate. D. The object is speeding up at an increasing rate. E. The object’s speed is changing at a steady rate. Motion with constant acceleration—Figures 2.15 and 2.17 Motion with constant positive acceleration results in steadily increasing velocity. The equations of motion under constant acceleration The pages leading to the top of page 51 follow the derivation of four equations of constant acceleration. They are shown at right. Special mention is made of these four equations because they will permeate our study of kinematics (linear and circular, too). Follow the steps in Problem-Solving Strategy 2.1 for any problem involving motion with constant acceleration. vx = vox + axt (x xo) = {(1/2)(vox + vx)}t x = xo + voxt + 1/2axt2 vx2 = vox2 + 2ax(x xo) Use the equations to study motorcycle motion Refer to Example 2.4 and use the equations in a practical example illustrating a motorcycle and rider. (Page 51) Find the motorcycle position at t = 2.0 s. Where is the motorcyclist when his velocity is 25 m/s ? Study two bodies with different accelerations Refer to Example 2.5 and use the equations in a practical example illustrating a motorcycle and its rider chasing an SUV. (Page 52) How much time it will take before the officer catches up with the motorist? Free fall—Figure 2.22 A strobe light begins to fire as the apple is dropped. Notice how the space between images increases as the apple’s velocity grows. Constant acceleration g, due to gravity, is 9.8 m/s2 Free fall II—Example 2.6 Aristotle thought that heavier bodies would fall faster. Galileo is said to have dropped two objects, one light and one heavy, from the top of the Leaning Tower of Pisa to test his assertion that all bodies fall at the same rate. Astronaut Dave Scott tested this himself by dropping a hammer and a feather on the moon. Free fall II—Example 2.6 Page 54 A coin is dropped from the leaning Tower, starting from rest. Calculate its position and velocity after 1.0 s, 2.0 s and 3.0 s. Free fall III—Figure 2.24 Refer to Example 2.7. (Page 55) Initial upward speed is 15.0 m/s. (b) the velocity when the ball is 5.0 m above the railing. (c) the maximum height reached and the time at which it is reached. (d) the acceleration of the ball when it is at its maximum height. Is velocity zero at the highest point?—Figure 2.25 Free fall III—Example 2.8 Find the time when the ball in the previous problem is 5.00 m below roof railing. (Page 55)