Two small spheres of equal mass, and heading towards each other with equal speeds, undergo a head on collision `(` external force acts on system of two spheres `)`. Then which of the following statement is correct ?

Options

A. Their final velocities must be zero
B. Their final velocities may be zero
C. Each must have a final velocity equal to the other's intial velocity.
D. Their velocities must be reduced in magnitude.

Explanation

To solve the problem, we need to analyze the situation of two small spheres of equal mass colliding head-on with equal speeds. Let's break down the steps:

### Step 1: Understanding the Initial Conditions
We have two spheres, A and B, each with mass \( m \). They are moving towards each other with equal speeds \( v \). Therefore, the initial velocities can be represented as:
- Velocity of sphere A, \( v_A = +v \)
- Velocity of sphere B, \( v_B = -v \)

### Step 2: Considering the Collision
Since the spheres are colliding head-on, we need to consider the nature of the collision. The problem states that an external force acts on the system, but it does not specify the nature of the collision (elastic or inelastic).

### Step 3: Analyzing the Coefficient of Restitution
The coefficient of restitution (COR) is a measure of how elastic a collision is. It is defined as the ratio of the relative speed after the collision to the relative speed before the collision. The value of COR can range from 0 (perfectly inelastic collision) to 1 (perfectly elastic collision). Since the problem does not specify the COR, we will consider both possibilities.

### Step 4: Evaluating the Statements
Now, let's evaluate the provided statements based on our understanding:

1. **Their final velocities must be 0.**
- This is not necessarily true. The spheres can have different final velocities depending on the nature of the collision.

2. **Their final velocities may be 0.**
- This is true. In a perfectly elastic collision, it is possible for both spheres to come to rest after the collision if they collide with equal and opposite momentum.

3. **Each must have a final velocity equal to the other's initial velocity.**
- This is not true. In an elastic collision, they can exchange velocities, but it is not a requirement that they must equal the initial velocities of the other sphere.

4. **Their velocity must be reduced in magnitude.**
- This is not necessarily true. Depending on the nature of the collision, their velocities could remain the same or change in magnitude.

### Conclusion
The correct statement among the options provided is: **"Their final velocities may be 0."**