A bullet of mass m is fired with velocity v in a large bulk of mass M. The final velocity v' will be

Options

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

Explanation

To solve the problem of a bullet of mass \( m \) fired with velocity \( v \) into a large bulk of mass \( M \), we can use the principle of conservation of momentum. Here’s a step-by-step solution:

### Step 1: Understand the system
We have a bullet of mass \( m \) moving with an initial velocity \( v \) and a large bulk of mass \( M \) which is initially at rest. When the bullet embeds itself into the bulk, we need to find the final velocity \( v' \) of the combined system.

### Step 2: Write down the conservation of momentum
According to the law of conservation of momentum, the total momentum before the event (the bullet being fired) must equal the total momentum after the event (the bullet embedded in the bulk).

**Initial momentum**:
- The bullet has momentum given by \( p_{\text{initial}} = m \cdot v \).
- The bulk is at rest, so its momentum is \( 0 \).

Thus, the total initial momentum is:
\[ p_{\text{initial}} = m \cdot v + 0 = m \cdot v \]

**Final momentum**:
- After the bullet embeds itself in the bulk, the total mass of the system becomes \( M + m \) and moves with a final velocity \( v' \).
- Therefore, the final momentum is given by \( p_{\text{final}} = (M + m) \cdot v' \).

### Step 3: Set initial momentum equal to final momentum
Using the conservation of momentum:
\[ m \cdot v = (M + m) \cdot v' \]

### Step 4: Solve for the final velocity \( v' \)
We can rearrange the equation to solve for \( v' \):
\[ v' = \frac{m \cdot v}{M + m} \]

### Conclusion
The final velocity \( v' \) of the combined system (the bullet and the bulk) after the bullet is fired into the bulk is:
\[ v' = \frac{m \cdot v}{M + m} \]