A sphere P of mass m and velocity `v_i` undergoes an oblique and perfectly elastic collision with an identical sphere Q initially at rest. The angle `theta` between the velocities of the spheres after the collision shall be

Options

A. 0
B. `45^@`
C. `90^@`
D. `180^@`

Explanation

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

### Step 1: Understand the scenario
We have two identical spheres, P and Q. Sphere P has an initial velocity \( v_i \) and sphere Q is initially at rest. After the collision, we need to find the angle \( \theta \) between their velocities.

### Step 2: Apply the conservation of momentum
In a perfectly elastic collision, both momentum and kinetic energy are conserved. The conservation of momentum can be expressed as:
\[ m v_i = m v_{pf} + m v_{qf} \]
where \( v_{pf} \) and \( v_{qf} \) are the final velocities of spheres P and Q, respectively. Since the masses are equal, we can simplify this to:
\[ v_i = v_{pf} + v_{qf} \]

### Step 3: Apply the conservation of kinetic energy
The conservation of kinetic energy states that the total kinetic energy before the collision is equal to the total kinetic energy after the collision:
\[ \frac{1}{2} m v_i^2 = \frac{1}{2} m v_{pf}^2 + \frac{1}{2} m v_{qf}^2 \]
Again, since the masses are equal, we can cancel \( \frac{1}{2} m \) from all terms:
\[ v_i^2 = v_{pf}^2 + v_{qf}^2 \]

### Step 4: Relate the equations
From the conservation of momentum, we have:
\[ v_{pf} + v_{qf} = v_i \]
Squaring both sides gives:
\[ (v_{pf} + v_{qf})^2 = v_i^2 \]
Expanding this, we get:
\[ v_{pf}^2 + 2v_{pf}v_{qf} + v_{qf}^2 = v_i^2 \]

### Step 5: Substitute the kinetic energy equation
Now, we can substitute the kinetic energy equation \( v_i^2 = v_{pf}^2 + v_{qf}^2 \) into the expanded momentum equation:
\[ v_{pf}^2 + 2v_{pf}v_{qf} + v_{qf}^2 = v_{pf}^2 + v_{qf}^2 \]
This simplifies to:
\[ 2v_{pf}v_{qf} = 0 \]
This implies that either \( v_{pf} = 0 \) or \( v_{qf} = 0 \), which is not possible since both spheres are in motion after the collision.

### Step 6: Determine the angle \( \theta \)
Since \( 2v_{pf}v_{qf} = 0 \) indicates that the two velocities are perpendicular, we conclude that:
\[ \cos(\theta) = 0 \implies \theta = 90^\circ \]

### Final Answer
The angle \( \theta \) between the velocities of the spheres after the collision is:
\[ \theta = 90^\circ \]