A ball moving with velocity 2 m/s collides head on with another stationary ball of double the mass. If the coefficient of restitution is 0.5, then their velocities (in m/s) after collision will be

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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