A ball moving with velocity of 6 m/s strikes an identical stationary ball. After collision each ball moves at an angle of `30^(@)` with the original line of motion. What are the speeds of the balls after the collision ?

Options

A. `sqrt3/2 m /sec`
B. `3 m / sec`
C. `2 sqrt 3 m/sec`
D. `sqrt 3 m/sec`

Explanation

To solve the problem, we will use the principle of conservation of momentum. We have two identical balls, one moving and the other stationary. After the collision, both balls move at an angle of 30 degrees with respect to the original line of motion.

### Step-by-Step Solution:

1. **Identify Initial Conditions:**
- Let the mass of each ball be \( m \).
- The initial velocity of ball 1 (moving ball) is \( u_1 = 6 \, \text{m/s} \).
- The initial velocity of ball 2 (stationary ball) is \( u_2 = 0 \, \text{m/s} \).

2. **Identify Final Conditions:**
- After the collision, let the speeds of both balls be \( v_1 \) and \( v_2 \).
- Both balls move at an angle of \( 30^\circ \) with respect to the original line of motion.

3. **Apply Conservation of Momentum:**
- Momentum must be conserved in both the x-direction and y-direction.

4. **Momentum in the Y-Direction:**
- Initially, there is no momentum in the y-direction:
\[ 0 = m v_1 \sin(30^\circ) - m v_2 \sin(30^\circ) \]
- Simplifying, we get:
\[ v_1 \sin(30^\circ) = v_2 \sin(30^\circ) \]
- Since \( \sin(30^\circ) = \frac{1}{2} \), we can cancel it out:
\[ v_1 = v_2 \]
- Let \( v_1 = v_2 = v \).

5. **Momentum in the X-Direction:**
- The initial momentum in the x-direction is:
\[ m u_1 = m \cdot 6 \]
- The final momentum in the x-direction is:
\[ m v_1 \cos(30^\circ) + m v_2 \cos(30^\circ) \]
- Substituting \( v_1 = v_2 = v \):
\[ m \cdot 6 = m v \cos(30^\circ) + m v \cos(30^\circ) \]
- This simplifies to:
\[ 6 = 2v \cos(30^\circ) \]
- Since \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \):
\[ 6 = 2v \cdot \frac{\sqrt{3}}{2} \]
- Simplifying further:
\[ 6 = v \sqrt{3} \]
- Solving for \( v \):
\[ v = \frac{6}{\sqrt{3}} = 2\sqrt{3} \, \text{m/s} \]

6. **Final Speeds:**
- The speeds of both balls after the collision are:
\[ v_1 = v_2 = 2\sqrt{3} \, \text{m/s} \]

### Final Answer:
The speeds of both balls after the collision are \( 2\sqrt{3} \, \text{m/s} \).