C

Step 1: Consider two spheres, one with mass m moving with velocity u, and the other at rest. The coefficient of restitution (e) is defined as the relative speed of separation to the relative speed of approach.
Step 2: Before the collision, the relative velocity of approach is u - 0 = u.
Step 3: Let v1 be the velocity of sphere 1 (initially moving) after the collision and v2 be the velocity of sphere 2 (initially at rest) after the collision. By the definition of coefficient of restitution, we have:
$$e = \frac{v2 - v1}{u - 0}$$
Step 4: Rearranging gives us:
$$v2 - v1 = eu$$
Step 5: Applying conservation of momentum:
$$mu + 0 = mv1 + mv2$$

Simplifying leads to:
$$u = v1 + v2$$
Step 6: Now we have two equations:
1. $$v2 - v1 = eu$$
2. $$v1 + v2 = u$$
Step 7: From equation 1, we can express v2 as:
$$v2 = v1 + eu$$
Step 8: Substitute this into the momentum equation:
$$v1 + (v1 + eu) = u$$
This simplifies to
$$2v1 + eu = u$$
Step 9: Solving for v1 gives:
$$v1 = \frac{u - eu}{2} = \frac{u(1 - e)}{2}$$
Step 10: Now substituting v1 back into the equation for v2 gives:
$$v2 = v1 + eu = \frac{u(1 - e)}{2} + eu = \frac{u(1 - e + 2e)}{2} = \frac{u(1 + e)}{2}$$
Step 11: The ratio of the velocities is:
$$\frac{v1}{v2} = \frac{\frac{u(1 - e)}{2}}{\frac{u(1 + e)}{2}} = \frac{(1 - e)}{(1 + e)}$$
Therefore, the correct ratio is given by Option C: $$\frac{(1 - e)}{(1 + e)}.$$