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 will use the principles of conservation of momentum and the properties of elastic collisions.

### Step 1: Understand the scenario
We have two particles:
- Particle 1 (mass \( m_1 \)) is moving with an initial velocity \( V \).
- Particle 2 (mass \( m_2 \)) is initially at rest (velocity = 0).

After the collision:
- Particle 1 moves with a velocity of \( \frac{V}{2} \).
- Particle 2 moves with an unknown velocity \( V' \).

### Step 2: Apply the conservation of momentum
The total momentum before the collision must equal the total momentum after the collision.

**Before collision:**
\[ \text{Initial momentum} = m_1 \cdot V + m_2 \cdot 0 = m_1 V \]

**After collision:**
\[ \text{Final momentum} = m_1 \cdot \frac{V}{2} + m_2 \cdot V' \]

Setting the initial momentum equal to the final momentum:
\[ m_1 V = m_1 \cdot \frac{V}{2} + m_2 \cdot V' \]

### Step 3: Rearranging the momentum equation
We can rearrange the equation to solve for \( V' \):
\[ m_1 V - m_1 \cdot \frac{V}{2} = m_2 \cdot V' \]
\[ m_1 \left(V - \frac{V}{2}\right) = m_2 \cdot V' \]
\[ m_1 \cdot \frac{V}{2} = m_2 \cdot V' \]

### Step 4: Express \( V' \) in terms of \( V \)
Now, we can express \( V' \):
\[ V' = \frac{m_1}{m_2} \cdot \frac{V}{2} \]

### Step 5: Use the elastic collision property
For elastic collisions, the coefficient of restitution \( e \) is defined as:
\[ e = \frac{\text{Relative velocity after collision}}{\text{Relative velocity before collision}} \]
Given that the collision is elastic, \( e = 1 \).

The relative velocity before the collision is:
\[ V - 0 = V \]

The relative velocity after the collision is:
\[ V' - \frac{V}{2} \]

Setting up the equation:
\[ 1 = \frac{V' - \frac{V}{2}}{V} \]

### Step 6: Solve for \( V' \)
Rearranging gives:
\[ V' - \frac{V}{2} = V \]
\[ V' = V + \frac{V}{2} \]
\[ V' = \frac{3V}{2} \]

### Conclusion
The velocity of the second particle after the collision is:
\[ V' = \frac{3V}{2} \]