To solve for the velocities of the balls after collision, we apply the principles of conservation of momentum and the coefficient of restitution.

Data given are as follows:

– Mass of ball 1 is m₁ = m
Velocity of ball 1 is u₁ = 2 m/s
Mass of ball 2 is m₂ = 2m (it is double in mass compared to ball 1)
Velocity of ball 2 is u₂ = 0 m/s (ball is at rest)
Coefficient of restitution is e = 0.5

Step 1: Apply the principle of conservation of momentum

The total momentum before the collision is equal to the total momentum after the collision:

m₁ * u₁ + m₂ * u₂ = m₁ * v₁ + m₂ * v₂

Substitute the given values:

m * 2 + 2m * 0 = m * v₁ + 2m * v₂
2m = m * v₁ + 2m * v₂

Divide both sides by m:

2 = v₁ + 2v₂ (Equation 1)

Step 2: Utilization of the coefficient of restitution

Coefficient of restitution is defined by the following equation:
e = (relative speed after collision) / (relative speed before collision)
In this case, it can be derived as
e = (v₂ – v₁) / (u₁ – u₂)
Using the values:
0.5 = (v₂ – v₁) / (2 – 0)
0.5 = (v₂ – v₁) / 2
Multiply both sides by 2
1 = v₂ – v₁ Equation 2
Step 3: Solution of equations

We have two equations
1. v₁ + 2v₂ = 2
2. v₂ – v₁ = 1
From the last equation we may write
v₂ = v₁ + 1
This must be put in the first:
v₁ + 2v₁ + 2 = 2
3v₁ + 2 = 2
3v₁ = 0
v₁ = 0
Inserting the latter into the last of our initial equations:
v₂ = 0 + 1
v₂ = 1

Final velocities after collision:
Velocity of ball 1 (v₁) = 0 m/s
Velocity of ball 2 (v₂) = 1 m/s

Final Answer:
The final velocities after the collision will be 0 m/s, 1 m/s.

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