To analyze the problem of two balls P and Q in a frictionless circular groove, we need to consider the principles of momentum and the nature of elastic collisions. When ball P is projected towards ball Q, it will collide with it after a time T. The key here is to understand how the velocities change after the collision and how that affects the timing of the second strike.

Understanding the Initial Conditions

Initially, ball P is moving with a certain velocity \( v_0 \) towards ball Q, which is stationary. When P strikes Q, we can apply the principles of conservation of momentum and the coefficient of restitution (e) to determine their velocities after the collision.

Collision Dynamics

During the first collision, the velocities of the balls change according to the following equations:

• Let the final velocity of ball P after the collision be \( v_P' \).
• Let the final velocity of ball Q after the collision be \( v_Q' \).

Using the coefficient of restitution, we have:

• \( v_Q' = e \cdot v_0 \) (since Q was initially at rest)
• \( v_P' = (1 - e) \cdot v_0 \)

Calculating the Time for the Second Strike

After the first collision, both balls will move along the groove. Ball Q will move away from P with a velocity of \( e \cdot v_0 \), while ball P will move back towards Q with a reduced velocity of \( (1 - e) \cdot v_0 \).

Distance and Time Analysis

The distance between the two balls at the moment of the first collision is the diameter of the groove, which we can denote as \( D \). After the first collision, the time taken for the second collision can be calculated by considering the relative velocities of the two balls:

• The distance to be covered for the second strike is still \( D \).
• The relative velocity between the two balls after the first collision is \( v_P' + v_Q' = (1 - e)v_0 + e v_0 = v_0 \).

Now, the time \( t \) taken for the second strike can be calculated as:

• \( t = \frac{D}{v_P' + v_Q'} = \frac{D}{v_0} \)

Since the first collision occurred at time \( T \), the total time until the second strike is:

• \( T + t = T + \frac{D}{v_0} \)

Final Expression for Time Until Second Strike

Given that the time for the second strike is proportional to the initial velocity of ball P and the coefficient of restitution, we can express the total time until the second strike as:

• \( T + \frac{D}{v_0} = T + \frac{D}{v_0} \cdot \frac{1}{e} \) (since \( D \) is proportional to \( e \))

After simplifying, we find that the second strike occurs at:

• \( \frac{2T}{e} \)

Conclusion

Thus, the correct answer to the problem is option (1) \( \frac{2T}{e} \). This result illustrates how the dynamics of motion and collisions can be analyzed using fundamental principles of physics, leading to a clear understanding of the timing of events in a system.