Two billiard balls A and B each of mass 50 g and moving in opposite directions with speed of `5 m//s` each, collide and rebound with the same speed if the collision lasts for `10^(-3)` s which of the following statement(s) is (are) true?

Options

A. The impulse imparted to each ball is 0.25 kg `m//s` and te force on each ball is 250N
B. The impulse imparted to each ball is 0.25 kg `m//s` and the force exerted on each ball is `25xx10^(-5)N`
C. The impulse imparted to each ball is 0.5 N
D. The impulse and the force on each ball are equal in magnitude and opposite in direction

Explanation

To solve the problem, we need to analyze the collision of the two billiard balls A and B, each with a mass of 50 grams and moving in opposite directions at a speed of 5 m/s. The collision lasts for \(10^{-3}\) seconds, and we need to determine the validity of the given statements regarding the collision.

### Step-by-Step Solution:

1. **Convert Mass to Kilograms**:
The mass of each billiard ball is given as 50 grams. We need to convert this into kilograms for standard SI units.
\[ m = 50 \text{ g} = \frac{50}{1000} \text{ kg} = 0.05 \text{ kg} \]

**Hint**: Always convert mass into kilograms when using SI units.

2. **Determine Initial and Final Velocities**:
- For ball A: Initial velocity \(u_A = 5 \text{ m/s}\) (to the right), Final velocity \(v_A = -5 \text{ m/s}\) (to the left after collision).
- For ball B: Initial velocity \(u_B = -5 \text{ m/s}\) (to the left), Final velocity \(v_B = 5 \text{ m/s}\) (to the right after collision).

**Hint**: Remember that direction matters; assign positive and negative signs based on the chosen reference direction.

3. **Calculate Change in Momentum for Each Ball**:
The change in momentum (\(\Delta p\)) for each ball can be calculated using the formula:
\[ \Delta p = m(v - u) \]
For ball A:
\[ \Delta p_A = 0.05 \text{ kg} \times (-5 \text{ m/s} - 5 \text{ m/s}) = 0.05 \text{ kg} \times (-10 \text{ m/s}) = -0.5 \text{ kg m/s} \]
For ball B:
\[ \Delta p_B = 0.05 \text{ kg} \times (5 \text{ m/s} - (-5 \text{ m/s})) = 0.05 \text{ kg} \times (10 \text{ m/s}) = 0.5 \text{ kg m/s} \]

**Hint**: The change in momentum is equal in magnitude but opposite in direction for the two colliding objects.

4. **Calculate the Impulse**:
Impulse is equal to the change in momentum:
\[ \text{Impulse} = \Delta p = F \cdot t \]
We can find the average force exerted during the collision. Since the collision lasts for \(10^{-3}\) seconds:
\[ F \cdot 10^{-3} = 0.5 \text{ kg m/s} \]
\[ F = \frac{0.5 \text{ kg m/s}}{10^{-3} \text{ s}} = 500 \text{ N} \]

**Hint**: Impulse can also be calculated using the time duration of the collision and the force applied.

5. **Conclusion**:
- The force exerted on both balls is equal in magnitude but opposite in direction, which confirms that the impulse and force are equal and opposite.
- Therefore, the correct statements regarding the collision are that the impulse and force on both balls will be equal and opposite in magnitude.

### Final Statements:
- The option stating that the impulse and force will be equal and opposite in magnitude is true.
- The option stating that 0.5 Newton is impacting is false as the calculated force is 500 Newtons.