Explanation:
When a ball impinges directly into another ball, the collision is said to be a one-dimensional collision. The coefficient of restitution (e) is defined as the ratio of relative velocity of separation to the relative velocity of approach.

Formula: e = relative velocity of separation / relative velocity of approach

Given:
- The initial kinetic energy is halved after the collision (i.e, 1/2 of the initial kinetic energy is lost).
- One ball is at rest before the collision.

Solution:
Let us assume that the mass of the moving ball is m and its initial velocity is v. The mass of the stationary ball is M.

Step 1:
The total initial kinetic energy of the system is given by:
KE = (1/2)mv^2

Step 2:
After the collision, the moving ball comes to rest momentarily as it transfers all its kinetic energy to the stationary ball. Therefore, the total kinetic energy after the collision is given by:
KE' = (1/2)Mv^2

Step 3:
From the conservation of momentum, we can write:
mv = Mv'

where v' is the velocity of the two balls after the collision.

Step 4:
Using the above equation, we can write the relative velocity of approach as:
v_rel = v - v' = v - (mv/M)

Step 5:
The relative velocity of separation is zero as the stationary ball is at rest after the collision.

Step 6:
Using the formula for coefficient of restitution, we get:
e = v_rel / v_app = (v - mv/M) / v

Step 7:
Substituting KE and KE' in Step 1 and Step 2 respectively, we get:
(1/2)mv^2 = (1/2)Mv^2
=> m = M

Step 8:
Using the above relationship between m and M, we can simplify the expression for coefficient of restitution obtained in Step 6:
e = (v - v/2) / v
=> e = 0.5

Therefore, the coefficient of restitution is 0.5.