Explanation:

When ball P collides head-on with ball Q, the momentum and kinetic energy of the system are conserved.

Momentum Conservation:

The momentum of an object is defined as the product of its mass and velocity. According to the law of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision.

Before the collision:
Momentum of ball P = mass of P * velocity of P
Momentum of ball Q = mass of Q * velocity of Q (since ball Q is at rest)

After the collision:
Momentum of ball P = mass of P * velocity of P' (velocity of P after collision)
Momentum of ball Q = mass of Q * velocity of Q' (velocity of Q after collision)

Since the mass of both balls is the same, we can equate the initial and final momentum equations:

mass of P * velocity of P = mass of P * velocity of P' + mass of Q * velocity of Q'

Kinetic Energy Conservation:

The kinetic energy of an object is defined as half the product of its mass and the square of its velocity. According to the law of conservation of kinetic energy, the total kinetic energy before the collision is equal to the total kinetic energy after the collision.

Before the collision:
Kinetic energy of ball P = 0.5 * mass of P * (velocity of P)^2
Kinetic energy of ball Q = 0.5 * mass of Q * (velocity of Q)^2 (since ball Q is at rest)

After the collision:
Kinetic energy of ball P = 0.5 * mass of P * (velocity of P')^2
Kinetic energy of ball Q = 0.5 * mass of Q * (velocity of Q')^2

Since the masses of both balls are the same, we can equate the initial and final kinetic energy equations:

0.5 * mass of P * (velocity of P)^2 = 0.5 * mass of P * (velocity of P')^2 + 0.5 * mass of Q * (velocity of Q')^2

Coefficient of Restitution:

The coefficient of restitution (e) is a measure of how elastic a collision is. It is defined as the ratio of the relative velocity of separation to the relative velocity of approach.

e = (velocity of separation) / (velocity of approach)

In this case, the coefficient of restitution is given as 1/4, which means:

e = 1/4

Since the collision is head-on, the relative velocity of approach is the sum of the velocities of the two balls before the collision, and the relative velocity of separation is the difference in their velocities after the collision.

velocity of approach = velocity of P - velocity of Q (since Q is at rest)
velocity of separation = velocity of P' - velocity of Q'

Therefore, we can write:

e = (velocity of P' - velocity of Q') / (velocity of P - velocity of Q)

Simplifying the equation:

1/4 = (velocity of P' - velocity of Q') / (velocity of P - velocity of Q)

Ratio of Velocities:

To find the ratio of the velocities of P and Q just after the collision, we can solve the above equation:

1/4 = (