A small 44 g ball strikes the ground at a speed of 28 m/s. It rebounds from the surface at a speed of 15 m/'s after being in contact for 3.2 ms. I. Approximate Evaluation First, we'll look at this problem from the point of view that the actual process of the bounce is not important, only the results. (a) What is the the average acceleration of the ball? ca> = 3.2 x 101-3r x k The coefficient of restitution is the ratio of the speed of a ball after a bounce to the speed before the bounce. Since the ball cannot have a negative speed, it is non-negative, and unless the bounce is powered, e.9. a pinball bumper, the ball will have a lower speed after the bounce, so the coefficient of restitution is typically less than one. (b) What is the coefficient of restitution in this case? r = .536 The simplest model for the ball's center of mass velocity during the bounce is given by the function *(t) = x(0) e-atcos(cot) k There are three parts to this equation: *(0) is the initial velocity, e"at describes the reduction in the speed with time, and cos(wt) describes how the ball is changing its direction of motion with time. The first part is a constant, while the other two change with time. [x is the Leibnizian notion the time rate of change of the position.] I s T/4 1 - T/2 II. Functional Evaluation First, we'll analyze the function itself given the data we have about the bounce. Essentially, we want to determine the three constants in the equation: the initial velocity cos(wt), the damping constant 1, and the angular frequency co. (a) From the data, what is the initial velocity of the ball? *(0) = 28 m/s X k (b) The damping term endt describes the reduction in the speed of the ball with time. What is the damping constant lambda that would match this problem? 1 = 2.37 s'-1 x [Hint: this is related to the coefficient of restitution.] (c) The elastic term cos(wt) describes how the ball changes direction. It starts the process with a value of 1, and it ends the process with a value of -1. What is the value of the angular frequency that governs this term? 0 = |625 rad/s x