Question:

A particle of mass m and zero initial speed is dropped from a height h above a horizontal table. The particle bounces vertically on the table. The coefficient of restitution for the impacts is e < 1. All motion is vertical, and each impact is considered instantaneous.

(i) How many times will the particle bounce before it ceases to bounce?
(ii) How long does it take before the particle ceases to bounce, measured from the moment of first impact?

Answer:

Step 1: First Impact

The speed just before the first impact is given by energy conservation:

After the first bounce, the upward speed becomes:

This gives a rebound height:

Step 2: General Case for Rebound Heights

After the n-th bounce, the height becomes:

The particle never completely stops bouncing, but the height becomes negligible as n → ∞. We define that the particle “ceases to bounce” when the height is less than a small threshold ε.

Taking logarithms:

Since e < 1, ln(e) < 0, the inequality flips. We choose the smallest integer n satisfying this condition.

Step 3: Time Calculation

Time to fall from height h:

Time to rise to height h₁ = e²h:

Total time of the first bounce loop (up + down):

Step 4: Time Series for All Bounces

Each subsequent bounce takes less time due to decreasing height. So the total time for all remaining bounces forms a geometric series:

Adding the time for the initial drop:

Final Results:

• Number of bounces: Infinite in theory, but practically ends when height < ε.
• Total time before bouncing stops: √(2h/g) × [(1 + 2e) / (1 – e)]

Conclusion:

The motion of a bouncing particle with energy loss on each impact can be modeled using geometric series. While the particle theoretically never stops bouncing due to continually halving rebound heights, we consider it stopped when the height falls below a practical threshold. The total time before it ceases to bounce can be calculated exactly using the coefficient of restitution and initial drop height.