A ball of mass 1 kg moving with a velocity of `"0.4 ms"^(-1)` collides with another stationary ball. After the collision, the first ball moves with a velocity of `0.3ms^(1)` in a direction making an angle of `90^(@)` with its initial direction. The momentum of the second ball after the collision will be (in kg `ms^(-1)`)

Options

A. 0.1
B. 0.3
C. 0.5
D. 0.7

Answer

To solve the problem, we will use the principle of conservation of momentum. The momentum before the collision must equal the momentum after the collision since there are no external forces acting on the system.

### Step-by-Step Solution:

1. **Identify Initial Momentum**:
- The first ball (mass \( m_1 = 1 \, \text{kg} \)) is moving with a velocity \( u_1 = 0.4 \, \text{m/s} \).
- The second ball is stationary, so its initial momentum is zero.
- The initial momentum of the system can be calculated as:
\[ p_{\text{initial}} = m_1 \cdot u_1 + m_2 \cdot 0 = 1 \cdot 0.4 + 0 = 0.4 \, \text{kg m/s} \, \text{(in the x-direction)} \]

2. **Identify Final Momentum**:
- After the collision, the first ball moves with a velocity \( v_1 = 0.3 \, \text{m/s} \) in a direction making an angle of \( 90^\circ \) with its initial direction. This means it moves in the y-direction.
- The momentum of the first ball after the collision is:
\[ p_{1,\text{final}} = m_1 \cdot v_1 = 1 \cdot 0.3 = 0.3 \, \text{kg m/s} \, \text{(in the y-direction)} \]

3. **Set Up the Conservation of Momentum Equation**:
- Let \( p_2 \) be the momentum of the second ball after the collision. According to the conservation of momentum:
\[ p_{\text{initial}} = p_{1,\text{final}} + p_{2,\text{final}} \]
- This can be expressed as:
\[ 0.4 \, \hat{i} = 0.3 \, \hat{j} + p_2 \]

4. **Solve for the Momentum of the Second Ball**:
- Rearranging the equation gives:
\[ p_2 = 0.4 \, \hat{i} - 0.3 \, \hat{j} \]

5. **Calculate the Magnitude of the Momentum of the Second Ball**:
- The magnitude of \( p_2 \) can be calculated using the Pythagorean theorem:
\[ |p_2| = \sqrt{(0.4)^2 + (-0.3)^2} = \sqrt{0.16 + 0.09} = \sqrt{0.25} = 0.5 \, \text{kg m/s} \]

### Final Answer:
The momentum of the second ball after the collision is \( 0.5 \, \text{kg m/s} \).

---