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. `v`
B. 2v
C. `(3)/(4)v`
D. `(4)/(3)v`

Answer

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

### Step 1: Understand the initial momentum
The initial momentum of the shell before it breaks can be calculated using the formula:
\[ \text{Initial Momentum} = \text{mass} \times \text{velocity} = m \times v \]

### Step 2: Identify the masses after the explosion
After the shell breaks, we have two pieces:
1. The first piece has a mass of \( \frac{m}{4} \) and remains stationary, so its momentum is:
\[ \text{Momentum of first piece} = \frac{m}{4} \times 0 = 0 \]
2. The second piece has a mass of \( \frac{3m}{4} \) and we need to find its velocity, which we will denote as \( u \).

### Step 3: Write the equation for the final momentum
The total momentum after the explosion must equal the total momentum before the explosion. Therefore, we can write:
\[ \text{Initial Momentum} = \text{Final Momentum} \]
This gives us:
\[ m \times v = 0 + \left(\frac{3m}{4}\right) \times u \]

### Step 4: Simplify the equation
Now we can simplify the equation:
\[ m \times v = \frac{3m}{4} \times u \]
Dividing both sides by \( m \) (assuming \( m \neq 0 \)):
\[ v = \frac{3}{4} u \]

### Step 5: Solve for \( u \)
To find \( u \), we rearrange the equation:
\[ u = \frac{4}{3} v \]

### Conclusion
The velocity of the second shell (the piece with mass \( \frac{3m}{4} \)) after the explosion is:
\[ u = \frac{4v}{3} \]