Consider an oblique elastic collision between a moving ball and a stationary ball of the same mass. Both the balls move with the same speed after the collision. After the collision, the angle between the directions of motion of two balls is x degree. Find the value of x.

Explanation

To solve the problem of finding the angle \( x \) between the directions of motion of two balls after an oblique elastic collision, we can follow these steps:

### Step 1: Understand the scenario
We have two balls of equal mass. One ball is moving with an initial velocity \( v_0 \) at an angle \( \alpha \) to the horizontal, while the other ball is stationary. After the collision, both balls move with the same speed.

### Step 2: Analyze the collision
Since the collision is elastic, both momentum and kinetic energy are conserved. We will analyze the velocities of the balls before and after the collision.

### Step 3: Break down the initial velocity
The initial velocity \( v_0 \) of the moving ball can be broken down into its components:
- Horizontal component: \( v_{0x} = v_0 \cos \alpha \)
- Vertical component: \( v_{0y} = v_0 \sin \alpha \)

### Step 4: Apply conservation of momentum
For an elastic collision, the momentum along the line of impact is conserved. Let \( v_1 \) and \( v_2 \) be the velocities of the two balls after the collision. The equations for conservation of momentum can be written as:
1. \( v_{0x} = v_1 + v_2 \) (along the line of impact)
2. The vertical component remains unchanged for the first ball: \( v_{0y} = v_{1y} \) (for the first ball).

### Step 5: Apply conservation of kinetic energy
The kinetic energy before and after the collision must also be equal:
\[ \frac{1}{2} m v_0^2 = \frac{1}{2} m v_1^2 + \frac{1}{2} m v_2^2 \]

### Step 6: Analyze the angles after collision
After the collision, the two balls move at an angle \( x \) to each other. Since they have the same speed after the collision, we can use trigonometric relationships to find the angle between their velocities.

### Step 7: Solve for the angle \( x \)
From the conservation of momentum and the fact that the two balls move with the same speed after the collision, we can deduce that the angle \( x \) between their paths must be \( 90^\circ \). This is because the momentum components must balance out in such a way that the two balls move perpendicularly to each other.

### Conclusion
Thus, the value of \( x \) is:
\[ x = 90^\circ \]

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