A ball is dropped from a height of 12 feet and bounces. Suppose that each bounce

Question

A ball is dropped from a height of 12 feet and bounces. Suppose that each bounce is 7/8 of the height of the bounce before. Thus, after the ball hits the floor for the first time, it rises to a height of 12(7/8) = 10.5 feet, etc. (Assume g = 32ft/s2 and no air resistance.) Find an expression for the height, in feet, to which the ball rises after it hits the floor for the nth time: hn = Find an expression for the total vertical distance the ball has traveled, in feet, when it hits the floor for the first, second, third and fourth times: first time: D = second time: D = third time: D = fourth time: D = Find an expression, in closed form, for the total vertical distance the ball has traveled when it hits the floor for the nth time. Dn=

Explanation / Answer

a)

after n bounces it will be 12 * (7/8)^n .

b) First TIme : 2*12*(7/8)

Second TIme : 2*12*(7/8)2

Third TIme : 2*12*(7/8)3

Fourth TIme : 2*12*(7/8)4

c) to calculate total distance travelled you can sum the geometric series sigma n = 1 to infinity 12 + 2* 12* 7/8 + 2* 12 * (7/8)^2 + 2 * 12 * (7/8)^3 + ... = 12 + 21( 1 + 7/8 + (7/8)^2 + ....)
= 12+ 21/(1-7/8) = 180 feet

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