A body of mass `m_(1)` moving with an unknown velocity of `v_(1)hat i`, undergoes a collinear collision with a body of mass `m_(2)` moving with a velocity `v_(2) hati`. After collision , `m_(1)` and `m_(2)` move with velocities of `v_(3) hati` and `V_(4) hati` , respectively. If `m_(2)=0.5 m_(1)` and `v_(3)=0.5v_(1)`, then `v_(1)` is :

Options

A. `v_(4)-v_(2)/2`
B. `v_(4)-V_(2)`
C. `V_(4)-V_(2)/4`
D. `v_(4)+v+_(2)`

Answer

To solve the problem, we will use the principle of conservation of linear momentum. Let's break down the solution step by step.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Mass of body 1: \( m_1 \)
- Mass of body 2: \( m_2 = 0.5 m_1 \)
- Initial velocity of body 1: \( v_1 \hat{i} \)
- Initial velocity of body 2: \( v_2 \hat{i} \)
- Final velocity of body 1: \( v_3 = 0.5 v_1 \hat{i} \)
- Final velocity of body 2: \( v_4 \hat{i} \)

2. **Write the Conservation of Momentum Equation:**
The conservation of momentum states that the total initial momentum equals the total final momentum. Therefore, we can write:
\[ m_1 v_1 + m_2 v_2 = m_1 v_3 + m_2 v_4 \]

3. **Substitute the Known Values:**
Substitute \( m_2 = 0.5 m_1 \) and \( v_3 = 0.5 v_1 \) into the equation:
\[ m_1 v_1 + 0.5 m_1 v_2 = m_1 (0.5 v_1) + 0.5 m_1 v_4 \]

4. **Factor Out \( m_1 \):**
Since \( m_1 \) is common on both sides, we can cancel it out (assuming \( m_1 \neq 0 \)):
\[ v_1 + 0.5 v_2 = 0.5 v_1 + 0.5 v_4 \]

5. **Rearrange the Equation:**
Move all terms involving \( v_1 \) to one side and the remaining terms to the other:
\[ v_1 - 0.5 v_1 = 0.5 v_4 - 0.5 v_2 \]
This simplifies to:
\[ 0.5 v_1 = 0.5 v_4 - 0.5 v_2 \]

6. **Multiply Through by 2:**
To eliminate the fraction, multiply the entire equation by 2:
\[ v_1 = v_4 - v_2 \]

### Final Answer:
Thus, the velocity \( v_1 \) is given by:
\[ v_1 = v_4 - v_2 \]