perplexus.info :: Just Math : Boing! Boing!

Boing! Boing! (Posted on 2019-05-16)

The equipment in our laboratory consists of an infinitely long inclined plane at 45 degrees to the vertical and a ball, which bounces with no loss in energy.

The ball is dropped from some height above the plane, bounces on its way down the plane (Boing! Boing!).

The laboratory investigator records:

T(N) = the time between the Nth and (N+1)th bounce, and

H(N) = the maximum distance between the ball and the plane between the Nth and (N+1)th bounce.

Write an expression for

H(N+1)T(N+1)/( H(N)T(N) ) as a function of N.

Submitted by FrankM

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Solution: (Hide)

1
As both H and T are independent of N
Explanation
It's easiest to reach the solution by considering an equivalent problem. First, consider a ball bouncing on flat ground. In this case, it is obvious that T and H are constants.
Now imagine that the ball is subject to a constant wind, blowing sideways, in addition to the effect of gravity. This gives a constant acceleration in two orthogonal directions. (Alternatively, we can do without the wind and consider the perspective of an observer travelling horizontally with constant acceleration). Since the accelerations are orthogonal, they do not interact. So H, T are constant in this case as well.
Now return to the original scenario, and consider the accelerations acting on the ball in two perpendicular directions: along the inclined plane and normal to the inclined plane. These accelerations are both constant values, namely g/sqrt(2). This is exactly the scenario investigated in the wind (or accelerating observer) scenarios.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Comment on attempt at better edition of soln text	FrankM	2019-05-27 09:09:22
attempt at better edition of soln text	Steven Lord	2019-05-17 13:25:30
soln	Steven Lord	2019-05-17 13:20:52

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