A shell of mass m moving with velocity v suddenly breaks into 2 pieces. The part having mass m /4 remains stationary. The velocity of the other shell will be

Options

A. u
B. 2u
C. `( 3u)/( 4)`
D. `( 4u)/( 3)`

Answer

To solve the problem, we will apply the principle of conservation of momentum. Here are the steps to find the velocity of the other piece after the shell breaks:

### Step-by-Step Solution:

1. **Identify the Initial Momentum:**
The initial momentum of the shell can be calculated using the formula:
\[ \text{Initial Momentum} (P_i) = m \cdot v \]
where \( m \) is the mass of the shell and \( v \) is its initial velocity.

2. **Analyze the Final State:**
After the shell breaks into two pieces:
- One piece has a mass of \( \frac{m}{4} \) and remains stationary, so its velocity is \( 0 \).
- The other piece has a mass of \( \frac{3m}{4} \) and we will denote its velocity as \( V_0 \).

3. **Calculate the Final Momentum:**
The final momentum of the system after the shell breaks can be expressed as:
\[ \text{Final Momentum} (P_f) = \left(\frac{3m}{4}\right) \cdot V_0 + \left(\frac{m}{4}\right) \cdot 0 \]
Simplifying this gives:
\[ P_f = \frac{3m}{4} \cdot V_0 \]

4. **Apply Conservation of Momentum:**
According to the law of conservation of momentum, the initial momentum must equal the final momentum:
\[ P_i = P_f \]
Thus, we have:
\[ m \cdot v = \frac{3m}{4} \cdot V_0 \]

5. **Solve for \( V_0 \):**
We can simplify the equation by canceling \( m \) from both sides (assuming \( m \neq 0 \)):
\[ v = \frac{3}{4} V_0 \]
Rearranging this equation to solve for \( V_0 \):
\[ V_0 = \frac{4}{3} v \]

6. **Conclusion:**
The velocity of the other piece (the one with mass \( \frac{3m}{4} \)) after the shell breaks is:
\[ V_0 = \frac{4}{3} v \]
This velocity is in the same direction as the initial velocity \( v \).