Coefficient of restitution – expression for collinear impact (same direction)\nTwo bodies move in the same straight line with initial velocities u1 and u2 (u1 > u2) before impact, and velocities v1 and v2 after impact, along the same direction. The coefficient of restitution e is:

Introduction / Context:
The coefficient of restitution (e) quantifies how ‘‘bouncy’’ a collision is along the line of impact. It is defined as the ratio of the relative speed of separation to the relative speed of approach.

Given Data / Assumptions:

• One-dimensional (collinear) impact.
• Before impact: velocities u1 and u2 in the same direction, with u1 > u2 so that collision occurs.
• After impact: velocities v1 and v2 in the same line.
• Standard sign convention: speeds taken along the same positive direction.

Concept / Approach:
By definition, e = (relative velocity of separation) / (relative velocity of approach), taken along the line of impact with consistent direction signs.

Step-by-Step Solution:

Verification / Alternative check:
Check dimensions: the ratio is dimensionless as required. Test the limits: if the bodies stick (v1 = v2), numerator is zero → e = 0; if they exchange speeds in an elastic case, numerator equals denominator → e = 1.

Why Other Options Are Wrong:

• Signs reversed in (a) and (c) produce negative or inverted ratios.
• (d) uses sums of velocities, not relative speeds, violating the definition.
• ‘‘None’’ is incorrect since (b) is valid.

Common Pitfalls:
Mixing direction signs or using speeds instead of signed velocities; always apply the definition consistently along the line of impact.

Final Answer:
e = (v2 − v1) / (u1 − u2)