A particle strikes elastically with another particle with velocity V after collision its move with half the velocity in the same direction find the velocity of the second particle if it is initially at rest

Options

A. `(3V)/2`
B. `V/2`
C. V
D. none of these

Answer

To solve the problem step by step, we need to apply the principles of conservation of momentum and the characteristics of elastic collisions.

### Step-by-Step Solution:

1. **Understand the Problem**:
- We have two particles.
- Particle 1 (P1) is moving with an initial velocity \( V \).
- Particle 2 (P2) is initially at rest (velocity = 0).
- After the elastic collision, P1 moves with a velocity of \( \frac{V}{2} \).

2. **Define the Variables**:
- Let \( V_1 \) be the initial velocity of P1 = \( V \).
- Let \( V_2 \) be the initial velocity of P2 = 0.
- Let \( V_1' \) be the final velocity of P1 after the collision = \( \frac{V}{2} \).
- Let \( V_2' \) be the final velocity of P2 after the collision (which we need to find).

3. **Apply Conservation of Momentum**:
The total momentum before the collision must equal the total momentum after the collision.
\[ \text{Initial Momentum} = \text{Final Momentum} \]
\[ m_1 V_1 + m_2 V_2 = m_1 V_1' + m_2 V_2' \]
Assuming both particles have the same mass \( m \):
\[ mV + 0 = m\left(\frac{V}{2}\right) + mV_2' \]
Simplifying, we get:
\[ V = \frac{V}{2} + V_2' \]

4. **Solve for \( V_2' \)**:
Rearranging the equation gives:
\[ V_2' = V - \frac{V}{2} \]
\[ V_2' = \frac{V}{2} \]

5. **Use the Elastic Collision Equation**:
For an elastic collision, the relative velocity of separation is equal to the relative velocity of approach:
\[ V_2' - V_1' = V_1 - V_2 \]
Substituting the known values:
\[ V_2' - \frac{V}{2} = V - 0 \]
\[ V_2' - \frac{V}{2} = V \]
Rearranging gives:
\[ V_2' = V + \frac{V}{2} = \frac{3V}{2} \]

6. **Conclusion**:
The final velocity of the second particle (P2) after the collision is \( \frac{3V}{2} \).

### Final Answer:
The velocity of the second particle after the collision is \( \frac{3V}{2} \).

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