Question

If \( u_1 \) and \( u_2 \) are the velocities of two bodies moving in the same direction before impact and \( v_1 \) and \( v_2 \) are the velocities of the same two bodies after impact, then the coefficient of restitution is

• A. \( \dfrac{v_1 - v_2}{u_1 - u_2} \)
• B. \( \dfrac{v_2 - v_1}{u_1 - u_2} \)
• C. \( \dfrac{u_1 - u_2}{v_1 - v_2} \)
• D. \( \dfrac{u_2 - u_1}{v_1 - v_2} \)

Hint:

Always subtract velocities in the direction of motion while finding relative velocities in impact problems.

Answer 1

The Correct Option is B

Step 1: Definition of coefficient of restitution.
The coefficient of restitution \( e \) is defined as the ratio of the relative velocity of separation to the relative velocity of approach along the line of impact.
Step 2: Writing the mathematical expression.
Before impact, relative velocity of approach: \[ u_1 - u_2 \] After impact, relative velocity of separation: \[ v_2 - v_1 \]
Step 3: Formula substitution.
\[ e = \frac{\text{Relative velocity of separation}}{\text{Relative velocity of approach}} \] \[ e = \frac{v_2 - v_1}{u_1 - u_2} \]
Step 4: Conclusion.
The correct expression for the coefficient of restitution is \[ \dfrac{v_2 - v_1}{u_1 - u_2} \]

Answer 2

The coefficient of restitution \( e \) is defined physically as how much of the closing (approach) speed between two bodies is recovered as a separating speed after their collision. Rather than quoting the formula directly, build it from the physical requirement that \( e \) must be positive whenever the bodies genuinely separate after colliding, and check each option against that requirement.

1. Option \( \dfrac{v_1 - v_2}{u_1 - u_2} \): Before impact, since the first body is chasing the second (moving in the same direction), \( u_1>u_2 \) so the denominator \( u_1-u_2 \) is positive. After a typical collision the faster body slows and the slower body speeds up, so usually \( v_1<v_2 \) after separation, making the numerator \( v_1-v_2 \) negative; this would make \( e \) come out negative in the ordinary case, which contradicts \( e \) being a non-negative physical quantity, so this ordering is not the correct definition.
2. Option \( \dfrac{v_2 - v_1}{u_1 - u_2} \): With \( u_1>u_2 \) before impact (denominator positive) and, after a normal separating collision, \( v_2>v_1 \) (the previously slower body now moves faster than the previously faster one, or at least they no longer approach), the numerator \( v_2-v_1 \) is positive too, keeping \( e \) positive as physically expected; this ordering, separation speed over approach speed, correctly matches the physical definition of restitution.
3. Option \( \dfrac{u_1 - u_2}{v_1 - v_2} \): This inverts the intended ratio, placing the approach speed in the numerator and the separation speed in the denominator, which is backwards relative to the definition of \( e \) as separation-over-approach.
4. Option \( \dfrac{u_2 - u_1}{v_1 - v_2} \): This is also inverted (approach in the numerator instead of denominator) and additionally has the before-impact terms in reversed order, compounding the mismatch with the standard definition.

Only the ratio that places the after-impact separation speed over the before-impact approach speed, in the correct order that keeps \( e \) non-negative, matches the physical definition.

So the correct answer is \( \dfrac{v_2 - v_1}{u_1 - u_2} \).