A body is moving with a velocity v , breaks up into two equal parts. One of the part retraces back with velocity v . Then the velocity of the other part is

Options

A. v in forward direction
B. 3v in forward direction
C. v in backward direction
D. 3v in backward direction.

Answer

To solve the problem, we will use the principle of conservation of momentum.

### Step-by-step Solution:

1. **Identify Initial Momentum**:
The initial momentum of the body before it breaks up can be calculated using the formula:
\[ p_{initial} = m \cdot v \]
where \( m \) is the mass of the body and \( v \) is its initial velocity.

2. **Break the Body into Two Parts**:
When the body breaks into two equal parts, each part has a mass of \( \frac{m}{2} \).

3. **Determine the Momentum of Each Part**:
Let’s denote the velocity of the first part (which retraces back) as \( v_1 = -v \) (negative because it is moving in the opposite direction). Let the velocity of the second part be \( v_2 \).

4. **Apply Conservation of Momentum**:
According to the conservation of momentum, the total momentum before the breakup must equal the total momentum after the breakup:
\[ p_{initial} = p_{final} \]
Therefore:
\[ m \cdot v = \left(\frac{m}{2} \cdot (-v)\right) + \left(\frac{m}{2} \cdot v_2\right) \]

5. **Simplify the Equation**:
We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ v = -\frac{1}{2}v + \frac{1}{2}v_2 \]

6. **Rearranging the Equation**:
Multiply the entire equation by 2 to eliminate the fraction:
\[ 2v = -v + v_2 \]
Now, add \( v \) to both sides:
\[ 2v + v = v_2 \]
Thus:
\[ v_2 = 3v \]

### Final Answer:
The velocity of the other part is \( 3v \).