When two spheres of equal masses undergo glancing elastic collision with one of them at rest after collision they will move

Options

A. opposite to one another
B. in the same direction
C. together
D. at right angle to each other

Explanation

To solve the problem of two spheres of equal masses undergoing a glancing elastic collision, we can follow these steps:

### Step 1: Understand the scenario
We have two spheres, both with equal mass \( m \). One sphere is moving with an initial velocity \( u_1 \) and the other sphere is at rest, so its initial velocity \( u_2 = 0 \).

### Step 2: Apply the conservation of momentum
In an elastic collision, the total momentum before the collision is equal to the total momentum after the collision. Mathematically, this can be expressed as:
\[ m u_1 + m u_2 = m v_1 + m v_2 \]
Since \( u_2 = 0 \), this simplifies to:
\[ m u_1 = m v_1 + m v_2 \]
Dividing through by \( m \) gives:
\[ u_1 = v_1 + v_2 \]

### Step 3: Apply the conservation of kinetic energy
In an elastic collision, the total kinetic energy before the collision is equal to the total kinetic energy after the collision. This can be expressed as:
\[ \frac{1}{2} m u_1^2 + \frac{1}{2} m u_2^2 = \frac{1}{2} m v_1^2 + \frac{1}{2} m v_2^2 \]
Again, since \( u_2 = 0 \), this simplifies to:
\[ \frac{1}{2} m u_1^2 = \frac{1}{2} m v_1^2 + \frac{1}{2} m v_2^2 \]
Dividing through by \( \frac{1}{2} m \) gives:
\[ u_1^2 = v_1^2 + v_2^2 \]

### Step 4: Analyze the angles
In a glancing collision, the two spheres will move off at right angles to each other after the collision. Thus, we can denote the angle between their velocities as \( 90^\circ \). This means:
\[ v_1^2 + v_2^2 = (u_1)^2 \]
This confirms that the velocities \( v_1 \) and \( v_2 \) are perpendicular to each other.

### Conclusion
From the above analysis, we conclude that after the collision, the two spheres will move at right angles to each other, confirming the statement in the question.

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