Two balls each of mass 2kg (one at rest) undergo oblique collision is perfectly elastic, then the angle between their velocities after collision is

Options

A. 30°
B. 60°
C. 45°
D. 90°

Answer

To solve the problem of two balls undergoing a perfectly elastic oblique collision, we will follow these steps:

### Step 1: Understand the Collision
We have two balls, each with a mass of 2 kg. One ball is moving while the other is at rest. Since the collision is perfectly elastic, both momentum and kinetic energy will be conserved.

### Step 2: Define the Initial Conditions
Let’s denote:
- Ball 1 (moving) has an initial velocity \( \vec{u_1} \) (let's say it moves along the x-axis).
- Ball 2 (at rest) has an initial velocity \( \vec{u_2} = 0 \).

### Step 3: Apply Conservation of Momentum
In an elastic collision, the total momentum before and after the collision must be equal. The momentum conservation equation can be written as:
\[ m_1 \vec{u_1} + m_2 \vec{u_2} = m_1 \vec{v_1} + m_2 \vec{v_2} \]
Since both balls have the same mass (2 kg), we can simplify this equation.

### Step 4: Analyze the Velocities After Collision
After the collision, let:
- Ball 1 moves at an angle \( \theta \) with velocity \( \vec{v_1} \).
- Ball 2 moves at an angle \( \phi \) with velocity \( \vec{v_2} \).

### Step 5: Use the Coefficient of Restitution
For perfectly elastic collisions, the coefficient of restitution \( e = 1 \). The equation for the coefficient of restitution along the line of impact can be expressed as:
\[ e = \frac{|\vec{v_2} - \vec{v_1}|}{|\vec{u_1} - \vec{u_2}|} \]
Since \( \vec{u_2} = 0 \), this simplifies to:
\[ 1 = \frac{|\vec{v_2} - \vec{v_1}|}{|\vec{u_1}|} \]

### Step 6: Determine the Angle Between Their Velocities
In a perfectly elastic collision between two equal masses, the two balls will move off at right angles to each other after the collision. This means that the angle \( \theta \) between their velocities after the collision is \( 90^\circ \).

### Conclusion
Thus, the angle between the velocities of the two balls after the collision is \( 90^\circ \).