A moving sphere of mass m suffer a perfect elastic collision (not head on) with equally massive stationary sphere. After collision both fly off at angle `theta`, value of which is :

Options

A. 0
B. `pi`
C. indeterminate
D. `pi//2`

Answer

To solve the problem of finding the angle \(\theta\) after a perfectly elastic collision between two equally massive spheres, we can follow these steps:

### Step 1: Understand the Scenario
We have two spheres of equal mass \(m\). One sphere is moving with velocity \(V\) and the other is stationary. After the collision, both spheres move off at an angle \(\theta\).

### Step 2: Break Down the Velocity Components
For the moving sphere, we can break its velocity \(V\) into two components relative to the line of impact:
- Along the line of impact: \(V \cos \theta\)
- Perpendicular to the line of impact: \(V \sin \theta\)

### Step 3: Analyze the Collision
In a perfectly elastic collision:
- The component of velocity along the line of impact gets exchanged.
- The component of velocity perpendicular to the line of impact remains unchanged.

### Step 4: After the Collision
After the collision:
- The first sphere (initially moving) will have its velocity component along the line of impact as \(V \sin \theta\) and will continue to move with this component.
- The second sphere (initially stationary) will now have a velocity component along the line of impact as \(V \cos \theta\).

### Step 5: Determine the Angles
Since the two spheres are of equal mass and the collision is perfectly elastic, the angle between their velocities after the collision will be \(90^\circ\). This means:
- The angle made by the velocity of the first sphere with the line of impact is \(\theta\).
- The angle made by the velocity of the second sphere with the line of impact is \(90^\circ - \theta\).

### Step 6: Conclusion
From the geometry of the situation, we can conclude that the two angles formed by the velocities of the spheres after the collision are complementary. Thus, the value of \(\theta\) can be derived from the relationship of the angles formed.

### Final Result
The value of \(\theta\) after the collision is:
\[ \theta = 45^\circ \]