1. In this problem you will use your smartphone and a superball to measure g. You will make two measurements, (1) the coefficient of restitution e of a superball hitting the floor, and (2) the time AT elapsed between the first bounce of a superball and the n + 1st bounce. You get to choose n; n = 5 or 7 is enough, but you can explore. First let's work out the theory relating these quantities to g. (a) Let hg be the initial height from which the ball is dropped from rest, and hi is the maximum height after the first bounce. We will disregard air resistance in this problem. Let the speed just before the ball bounces be vy, and the speed just after the bounce be v;. Then e = v, /vy is the coefficient of restitution. Use energy conservation to show e = y/hy/hg. (b) Show that the time between the first collision and the second collision is Tl = 2'01 /g (c) Generalize your result in (b) and show that the time between the nth collision and the n+1st collision is T}, = 2e"vo/g, or Tp = 2v/2e"\\/ho/g. (d) The total time between the first bounce and the nth bounce is AT = T+ T, +...T,. Show that (1) (e) Solve for g to show that _ Jhee? [1e" 2 where AT is the total time between the first and last collision, and n is the number of collisions with the floor after the first collision