The correct answer is $\boxed{\text{C}}$.
The coefficient of restitution is a measure of how much energy is lost in an elastic collision. A coefficient of restitution of $\frac{1}{2}$ means that half of the energy is lost in the collision.
In this case, the masses of the two balls are in the ratio of 2 : 1 and their respective velocities are in the ratio of 1 : 2 but in opposite direction before impact. This means that the total momentum of the system is zero.
After the collision, the velocities of the balls will be in the same direction, but the velocities of the two balls will be in the ratio of 1 : 2. The velocity of the heavier ball will be half the original velocity, and the velocity of the lighter ball will be the original velocity.
The following is a more detailed explanation of each option:
- Option A: The original velocity in the same direction. This is not possible because the total momentum of the system is zero.
- Option B: Half the original velocity in the same direction. This is possible, but it is not the only possible outcome.
- Option C: Half the original velocity in the opposite direction. This is possible, and it is the only possible outcome if the coefficient of restitution is $\frac{1}{2}$.
- Option D: Original velocity in the opposite direction. This is not possible because the total momentum of the system is zero.