Solution:

Given,
Mass of the two spheres = m
Velocity of first sphere = v
Velocity of second sphere = 2v
Coefficient of restitution = e = 1/2

To find:
Loss of kinetic energy of the system in the collision

Explanation:
When two spheres collide, they interact with each other through a force, which can be either attractive or repulsive. In this case, as the spheres are moving in opposite directions, they will collide head-on, and the force between them will be repulsive. Due to this force, the velocities of the two spheres will change, and their kinetic energy will be transferred from one to the other.

The coefficient of restitution is defined as the ratio of the relative velocities of the two spheres after the collision to the relative velocities before the collision. In this case, e=1/2, which means that the relative velocity of the two spheres after the collision is half of their relative velocity before the collision.

Let us assume that the two spheres stick together after the collision. In this case, the velocity of the combined sphere will be (m*v + m*2v)/(2m) = 3v/2. The kinetic energy of the combined sphere after the collision will be (1/2)*m*(3v/2)^2 = (9/8)*m*v^2.

The loss of kinetic energy of the system in the collision can be calculated as the difference between the initial kinetic energy of the two spheres and the final kinetic energy of the combined sphere. The initial kinetic energy of the two spheres is (1/2)*m*v^2 + (1/2)*m*(2v)^2 = (5/2)*m*v^2. Therefore, the loss of kinetic energy of the system in the collision is:

(5/2)*m*v^2 - (9/8)*m*v^2 = (7/8)*m*v^2

Therefore, the loss of kinetic energy of the system in the collision is (7/8)*m*v^2.

Conclusion:
The loss of kinetic energy of the system in the collision is (7/8)*m*v^2. When two spheres collide, their kinetic energy is transferred from one to the other, and the total kinetic energy of the system is not conserved. The coefficient of restitution is a measure of how much kinetic energy is retained by the system after the collision.