A ball is dropped from a height \( h \) onto a horizontal surface. If the coefficient of restitution is \( e \), then the total time after which it comes to rest is ________.

Question

10-08-2024
Physics

High School

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A ball is dropped from a height \( h \) onto a horizontal surface. If the coefficient of restitution is \( e \), then the total time after which it comes to rest is ________.

Answer :

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ssprabhatsingh24 ssprabhatsingh24 · Answer 1 · 2023-11-03T18:33:20-04:00

Final answer:

The total time for a ball to come to rest after being dropped from height h with a coefficient of restitution e involves summing an infinite series of bounce times. Each bounce time is successively smaller, as the ball reaches a height of e^2 times the previous bounce height. This results in a converging geometric series giving the total time.

Explanation:

When a ball is dropped from a height h onto a horizontal surface with a coefficient of restitution e, the total time t it takes to come to rest involves an infinite series of bounces where each bounce reaches a height less than the previous bounce by a factor of e2. The time to fall the first height h is t0 = √(2h/g), and each subsequent time to rise to the peak and fall back down is multiplied by e. To calculate the total time until the ball comes to rest, we utilize the geometric series of the time for each bounce, because the process is repeating with a common ratio of e.

Consider a ball dropped from an initial height h0 of 1 meter with a coefficient of restitution e of 0.5 and gravity g of 9.8 m/s2, the ball comes to rest in 1.36 seconds. The series of times will form a converging geometric series that can be summed to find the total time. This involves summing an infinite series that converges to a finite number, yielding us the total time taken before it comes to rest.