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 in a perfectly inelastic collision, we can follow these steps:

### Step 1: Understand the scenario
We have two balls:
- Ball 1 (mass = M) is moving with speed v.
- Ball 2 (mass = m) is at rest.

### Step 2: Apply the conservation of momentum
In a perfectly inelastic collision, the two balls stick together after the collision. Therefore, we can apply the conservation of momentum before and after the collision.

The initial momentum of the system is:
\[ P_{\text{initial}} = M \cdot v + m \cdot 0 = Mv \]

After the collision, the two balls move together with a common velocity \( V_0 \). The total mass after the collision is \( M + m \), so the final momentum is:
\[ P_{\text{final}} = (M + m) \cdot V_0 \]

By conservation of momentum:
\[ P_{\text{initial}} = P_{\text{final}} \]
\[ Mv = (M + m) \cdot V_0 \]

### Step 3: Solve for the final velocity \( V_0 \)
Rearranging the equation gives us:
\[ V_0 = \frac{Mv}{M + m} \]

### Step 4: Calculate the impulse
Impulse is defined as the change in momentum. The impulse \( I \) imparted to the first ball can be calculated as:
\[ I = P_{\text{final}} - P_{\text{initial}} \]

The initial momentum of the first ball is:
\[ P_{\text{initial}} = Mv \]

The final momentum of the first ball after the collision (which is now part of the combined mass) is:
\[ P_{\text{final}} = M \cdot V_0 = M \cdot \frac{Mv}{M + m} = \frac{M^2v}{M + m} \]

Now, substituting these into the impulse formula:
\[ I = \frac{M^2v}{M + m} - Mv \]

### Step 5: Simplify the impulse expression
To simplify:
\[ I = \frac{M^2v}{M + m} - \frac{Mv(M + m)}{M + m} \]
\[ I = \frac{M^2v - Mv(M + m)}{M + m} \]
\[ I = \frac{M^2v - M^2v - Mmv}{M + m} \]
\[ I = \frac{-Mmv}{M + m} \]

Since impulse is a magnitude, we take the absolute value:
\[ I = \frac{Mm v}{M + m} \]

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