A particle of mass m moving with velocity `vecV` makes a head on elastic collision with another particle of same mass initially at rest. The velocity of the first particle after the collision will be

Options

A. `vecV`
B. `-vecV`
C. `-2vecV`
D. zero

Answer

To solve the problem of finding the velocity of the first particle after it makes a head-on elastic collision with another particle of the same mass that is initially at rest, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the System**:
- We have two particles, both with mass \( m \).
- The first particle is moving with an initial velocity \( \vec{V} = v \).
- The second particle is initially at rest, so its initial velocity \( \vec{V_2} = 0 \).

2. **Conservation of Momentum**:
- In an elastic collision, the total momentum before the collision equals the total momentum after the collision.
- Before the collision, the total momentum \( p_{\text{initial}} \) is:
\[ p_{\text{initial}} = mv + 0 = mv \]
- After the collision, let the velocity of the first particle be \( v_1 \) and the second particle be \( v_2 \). The total momentum after the collision \( p_{\text{final}} \) is:
\[ p_{\text{final}} = mv_1 + mv_2 \]
- Setting the initial momentum equal to the final momentum gives us:
\[ mv = mv_1 + mv_2 \]
- Dividing through by \( m \) (assuming \( m \neq 0 \)):
\[ v = v_1 + v_2 \quad \text{(1)} \]

3. **Coefficient of Restitution**:
- For elastic collisions, the coefficient of restitution \( e = 1 \).
- The coefficient of restitution is defined as the ratio of the relative velocity of separation to the relative velocity of approach:
\[ e = \frac{v_2 - v_1}{v} \]
- Since \( e = 1 \):
\[ 1 = \frac{v_2 - v_1}{v} \]
- Rearranging gives:
\[ v = v_2 - v_1 \quad \text{(2)} \]

4. **Solving the Equations**:
- Now we have two equations:
- From (1): \( v = v_1 + v_2 \)
- From (2): \( v = v_2 - v_1 \)
- We can add these two equations:
\[ v + v = v_1 + v_2 + v_2 - v_1 \]
\[ 2v = 2v_2 \]
- Thus, we find:
\[ v_2 = v \]

5. **Finding \( v_1 \)**:
- Now substituting \( v_2 = v \) back into equation (1):
\[ v = v_1 + v \]
- This simplifies to:
\[ v_1 = 0 \]

### Conclusion:
The velocity of the first particle after the collision is \( v_1 = 0 \).