Two balls of masses m and 2m moving in opposite directions collide head on elastically with velocities v and `2v`. Find their velocities after collision.

Explanation

To solve the problem of two balls colliding elastically, we will use the principles of conservation of momentum and the coefficient of restitution. Let's denote the masses and velocities as follows:

- Mass of ball 1 (m) = m, initial velocity (u1) = v (moving in the positive direction)
- Mass of ball 2 (2m) = 2m, initial velocity (u2) = -2v (moving in the negative direction)

### Step 1: Apply Conservation of Momentum
The total momentum before the collision must equal the total momentum after the collision.

The equation for the conservation of momentum is:
\[ m \cdot u_1 + 2m \cdot u_2 = m \cdot v_1 + 2m \cdot v_2 \]

Substituting the known values:
\[ m \cdot v + 2m \cdot (-2v) = m \cdot v_1 + 2m \cdot v_2 \]

This simplifies to:
\[ mv - 4mv = mv_1 + 2mv_2 \]
\[ -3mv = mv_1 + 2mv_2 \]

Dividing through by m (assuming m ≠ 0):
\[ -3v = v_1 + 2v_2 \]
(Equation 1)

### Step 2: Apply the Coefficient of Restitution
For elastic collisions, the coefficient of restitution (e) is equal to 1. The coefficient of restitution is defined as:
\[ e = \frac{\text{Relative velocity of separation}}{\text{Relative velocity of approach}} \]

The relative velocity of approach is:
\[ u_1 - u_2 = v - (-2v) = v + 2v = 3v \]

The relative velocity of separation is:
\[ v_2 - v_1 \]

Setting up the equation using the coefficient of restitution:
\[ 1 = \frac{v_2 - v_1}{3v} \]

This simplifies to:
\[ v_2 - v_1 = 3v \]
(Equation 2)

### Step 3: Solve the Equations Simultaneously
Now we have two equations:
1. \( -3v = v_1 + 2v_2 \) (Equation 1)
2. \( v_2 - v_1 = 3v \) (Equation 2)

From Equation 2, we can express \( v_1 \) in terms of \( v_2 \):
\[ v_1 = v_2 - 3v \]

Substituting this expression for \( v_1 \) into Equation 1:
\[ -3v = (v_2 - 3v) + 2v_2 \]
\[ -3v = v_2 - 3v + 2v_2 \]
\[ -3v = 3v_2 - 3v \]

Adding \( 3v \) to both sides:
\[ 0 = 3v_2 \]

Thus, we find:
\[ v_2 = 0 \]

### Step 4: Find \( v_1 \)
Now substituting \( v_2 = 0 \) back into the expression for \( v_1 \):
\[ v_1 = 0 - 3v = -3v \]

### Final Velocities
- The final velocity of mass \( m \) (ball 1) is \( v_1 = -3v \) (moving in the negative direction).
- The final velocity of mass \( 2m \) (ball 2) is \( v_2 = 0 \) (it comes to rest).

### Summary of Results
- Final velocity of mass \( m \) (ball 1): \( -3v \)
- Final velocity of mass \( 2m \) (ball 2): \( 0 \)