Elastic collision with a wall — time ratio and coefficient of restitution: A smooth marble ball rolls on a horizontal floor toward a vertical wall. The time taken to return to the launch point is twice the time taken to reach the wall. What is the coefficient of restitution between the ball and the wall?

Introduction / Context:
Coefficient of restitution e relates the speeds of separation and approach in a one-dimensional impact. For a ball striking a fixed wall normally, e equals the ratio of rebound speed to approach speed. Timing information can be used to infer e when distances are equal and rolling resistance is negligible.

Given Data / Assumptions:

• Straight-line normal impact on a fixed wall.
• Horizontal floor, negligible rolling resistance and air drag.
• Outbound and inbound path lengths are equal.
• Return time is twice the outbound time.

Concept / Approach:

Let distance to the wall be D, approach speed be u, and rebound speed be v. Then time to reach the wall t1 = D / u; time to return t2 = D / v. Given t2 = 2 t1, we obtain v = u / 2. For a ball–wall impact, e = rebound speed / approach speed = v / u.

Step-by-Step Solution:

Verification / Alternative check:

If the ball took, say, 1 s to reach the wall, it would take 2 s to return, totaling 3 s, consistent with speeds u and u/2 on equal paths.

Why Other Options Are Wrong:

(a) and (c) correspond to v = 0.25 u or 0.75 u, contradicting the time ratio. (d) e = 1 implies perfectly elastic impact with equal times out and back. (e) would mean the ball sticks (no rebound), which is inconsistent.

Common Pitfalls:

Using distance ratio incorrectly; forgetting that for a fixed wall, e is speed ratio, not including sign; assuming oblique impact when the motion is normal.

Final Answer:

0.50