A ball of mass m moving at a speed v makes a head-on wllision with an identical ball at rest. The kinetic energy of the balls after the collision is 80% of the original. Find the coefficient of restilation.Correct answer is '0.77'. Can you explain this answer?

Question

Answer

Given:
- Mass of the balls: m
- Initial velocity of ball 1: v
- Final kinetic energy of the balls after collision: 80% of the original

To find:
- Coefficient of restitution

Explanation:

The coefficient of restitution (e) is a measure of the elasticity of a collision between two objects. It is defined as the ratio of the relative velocity of separation to the relative velocity of approach after the collision.

Step 1: Initial Kinetic Energy
Before the collision, the initial kinetic energy of the system can be calculated using the formula:

KE1 = (1/2)mv^2

Step 2: Final Kinetic Energy
After the collision, the final kinetic energy of the system is given as 80% of the initial kinetic energy:

KE2 = 0.8 * KE1

Step 3: Conservation of Momentum
Since the collision is head-on, the momentum is conserved. Therefore, the total momentum before the collision is equal to the total momentum after the collision. The momentum of an object is given by the product of its mass and velocity.

Initial momentum = mv
Final momentum of ball 1 = mv1
Final momentum of ball 2 = mv2

Since the second ball is initially at rest, its final velocity will be v2 = -v1.

Step 4: Using Kinetic Energy and Momentum Equations
Using the equations for kinetic energy and momentum, we can write:

KE1 = (1/2)mv^2
KE2 = (1/2)m(v1^2 + v2^2)
= (1/2)m(v1^2 + (-v1)^2)
= (1/2)m(2v1^2)
= mv1^2

Since KE2 = 0.8 * KE1, we can equate the two equations:

mv1^2 = 0.8 * (1/2)mv^2
v1^2 = 0.8 * v^2
v1 = √(0.8)v

Step 5: Coefficient of Restitution
The coefficient of restitution (e) can be calculated using the formula:

e = |v2 - v1| / |v|

Since v2 = -v1, we have:

e = |-v1 - v1| / |v|
= 2v1 / v
= 2√(0.8)v / v
= 2√(0.8)
≈ 0.77

Therefore, the coefficient of restitution is approximately 0.77.

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