Question

An object of mass 'm' moving with velocity 'u' collides with another stationary object of mass 'M' and stops just after the collision. The coefficient of restitution is ______.

• A. $\frac{m}{M + m}$
• B. $\frac{M - m}{M + m}$
• C. $\frac{m}{M}$
• D. 1

Hint:

If a perfectly elastic collision ($e=1$) occurs between identical masses ($m=M$), they exchange velocities. Here, $m$ stops and $M$ moves. If $m=M$, then $e=1$, which perfectly aligns with our derived formula $e=m/M$.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
A moving mass $m$ hits a stationary mass $M$. After the impact, mass $m$ comes to a complete halt, and mass $M$ moves forward. We need to find the coefficient of restitution ($e$) for this collision.

Step 2: Key Formula or Approach:
1. Conservation of Linear Momentum: Total initial momentum = Total final momentum.
$$m u_1 + M u_2 = m v_1 + M v_2$$
2. Coefficient of Restitution ($e$): The ratio of the relative velocity of separation to the relative velocity of approach.
$$e = \frac{v_2 - v_1}{u_1 - u_2}$$

Step 3: Detailed Explanation:
Let's list the known velocities:
Initial velocity of mass $m$: $u_1 = u$
Initial velocity of mass $M$: $u_2 = 0$ (stationary)
Final velocity of mass $m$: $v_1 = 0$ (stops)
Final velocity of mass $M$: $v_2 = V$ (unknown)
First, apply Conservation of Momentum to find $V$:
$$m(u) + M(0) = m(0) + M(V)$$
$$mu = MV \implies V = \frac{mu}{M}$$
Now, substitute these velocities into the coefficient of restitution formula:
$$e = \frac{v_2 - v_1}{u_1 - u_2}$$
$$e = \frac{V - 0}{u - 0} = \frac{V}{u}$$
Substitute the expression we found for $V$:
$$e = \frac{\frac{mu}{M}}{u}$$
The initial velocity $u$ cancels out:
$$e = \frac{m}{M}$$

Step 4: Final Answer:
The coefficient of restitution is $\frac{m}{M}$, matching option (c).