Coefficient of restitution from bounce heights A ball drops from 2.25 m on a smooth floor and rebounds to a height of 1.00 m. What is the coefficient of restitution e between the ball and the floor?

Introduction / Context:
The coefficient of restitution e quantifies collision elasticity along the line of impact. For vertical bounces under gravity with negligible air resistance, e can be found from pre- and post-impact speeds, which relate to drop and rebound heights via energy considerations.

Given Data / Assumptions:

• Drop height h1 = 2.25 m.
• Rebound height h2 = 1.00 m.
• Neglect air resistance; same impact point; smooth floor.

Concept / Approach:

Before impact, speed v1 just before hitting the floor satisfies v1^2 = 2 g h1. After impact, rebound speed v2 satisfies v2^2 = 2 g h2. By definition, e = v2 / v1 for a vertical bounce on a massive floor.

Step-by-Step Solution:

Verification / Alternative check:

Energy ratios for ideal vertical motion lead to the same relation e^2 = h2/h1, confirming the calculation.

Why Other Options Are Wrong:

0.25, 0.33, 0.50, and 0.75 do not match √(1/2.25). Only 0.67 (rounded) is consistent with the measured heights.

Common Pitfalls:

Using height ratio directly instead of its square root; mixing up drop and rebound heights; rounding too early.

Final Answer:

0.67