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 \[ \dfrac{v_2 - v_1}{u_1 - u_2} \]