Worksheet Average and Instantaneous Velocity Math 124 Introduction In this worksheet, we introduce what are called the average and instantaneous velocity in the context of a specific physical problem: A golf ball is hit toward the cup from a distance of 50 feet. Assume the distance from the ball to the cup at time t seconds is given by the function d(t) = 50?20t + 2t2. The graph of y = d(t) appears below. 1 2 3 4 5 6 10 20 30 40 50 1. Does the ball reach the cup? If so, when? (An- swer this question two ways: by using algebra, and by reading the graph.) . 2. (a) Plot and label these points on the graph: P = (2,d(2)) Q2 = (4,d(4)) Q1 = (3,d(3)) Q0.5 = (2.5,d(2.5)) Q0.01 = (2.01,d(2.01)) (b) Sketch the line through P and Q2 Sketch the line through P and Q1 Sketch the line through P and Q0.5 Sketch the line through P and Q0.01 (c) Compute the slopes of the lines in (b). (d) We define the average velocity as follows: average velocity=distance traveledtime elapsed . Explain why the slope of the line PQ2= the average velocity from t = 2 seconds to t = 4 seconds. (e) Find the average velocity over the following time intervals: t = 2 seconds to t = 4 seconds: t = 2 seconds to t = 3 seconds: t = 2 seconds to t = 2.5 seconds: t = 2 seconds to t = 2.01 seconds: (f) The average velocities in (e) approach a number as the time interval gets smaller and smaller. Guess this number. 3. Let h be a small constant positive number and define Qh = (2+h,d(2+h)). Compute the slope of the secant line connecting P and Qh by simplifying: slope = (ycoordinateQh)?(ycoordinateP)(tcoordinateQ h)?(tcoordinateP) so there is no h in the denominator. This slope = the average velocity on the time interval t = 2 seconds to t = 2 + h seconds. 4. What number do you get when you plug h = 0 into the simplified expression in problem 3 above? This is called the instantaneous velocity at t = 2 seconds. 5. Draw a line through P with slope equal to the number computed in the previous step. How would you describe this line relative to the graph?
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