A ball of mass M moving with speed v collides perfectly inelastically with another ball of mass m at rest. The magnitude of impulse imparted to the first ball is

Options

A. Mv
B. mv
C. `(Mm)/(M+m)v`
D. `(M^(2))/(M+m)v`

Answer

To solve the problem of finding the magnitude of impulse imparted to the first ball after a perfectly inelastic collision, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Initial Conditions:**
- Mass of the first ball (moving) = \( M \)
- Speed of the first ball = \( v \)
- Mass of the second ball (at rest) = \( m \)
- Speed of the second ball = \( 0 \)

2. **Calculate Initial Momentum:**
- The initial momentum of the system is given by the momentum of the first ball since the second ball is at rest.
\[ \text{Initial Momentum} = M \cdot v + m \cdot 0 = Mv \]

3. **Understand the Nature of Collision:**
- Since the collision is perfectly inelastic, the two balls stick together after the collision.

4. **Calculate Final Momentum:**
- After the collision, the combined mass of the two balls is \( M + m \).
- Let the final velocity of the combined mass be \( v' \).
- By conservation of momentum:
\[ \text{Initial Momentum} = \text{Final Momentum} \]
\[ Mv = (M + m)v' \]

5. **Solve for Final Velocity \( v' \):**
\[ v' = \frac{Mv}{M + m} \]

6. **Calculate Change in Momentum:**
- The change in momentum for the first ball is given by:
\[ \text{Change in Momentum} = \text{Final Momentum} - \text{Initial Momentum} \]
- The final momentum of the first ball after the collision is:
\[ \text{Final Momentum of first ball} = M \cdot v' \]
- Thus,
\[ \text{Change in Momentum} = M \cdot v' - M \cdot v \]
- Substituting \( v' \):
\[ \text{Change in Momentum} = M \left( \frac{Mv}{M + m} \right) - Mv \]

7. **Simplify the Change in Momentum:**
\[ = \frac{M^2v}{M + m} - \frac{Mv(M + m)}{M + m} \]
\[ = \frac{M^2v - Mv(M + m)}{M + m} \]
\[ = \frac{M^2v - M^2v - Mmv}{M + m} \]
\[ = \frac{-Mmv}{M + m} \]

8. **Magnitude of Impulse:**
- Impulse is equal to the change in momentum, hence:
\[ \text{Impulse} = \left| \frac{-Mmv}{M + m} \right| = \frac{Mmv}{M + m} \]

### Final Answer:
The magnitude of impulse imparted to the first ball is:
\[ \text{Impulse} = \frac{M \cdot m \cdot v}{M + m} \]