To tackle this problem, we need to break it down into manageable parts. We have two balls, m and M, on a frictionless circular track, separated by a spring. When the string holding them together breaks, the spring pushes the balls apart, and they eventually collide again. Let's analyze each part step by step.

Part (a): Collision Angle Calculation

When the string breaks, the spring pushes the two balls in opposite directions. Since the track is circular and frictionless, both balls will move along the circumference of the circle. The key here is to understand how far each ball travels before they collide again.

Let’s denote the angle through which ball M travels as θ. The distance each ball travels along the circular track can be expressed in terms of the radius R and the angle θ:

• Distance traveled by ball M: s_M = Rθ
• Distance traveled by ball m: s_m = R(2π - θ)

Since the spring is massless and the track is frictionless, the acceleration of each ball will depend on their respective masses. By applying Newton's second law, we can express the forces acting on each ball. The spring exerts equal and opposite forces on both balls, leading to their acceleration:

• Acceleration of ball M: a_M = k / M
• Acceleration of ball m: a_m = k / m

Using the relationship between distance, acceleration, and time, we can derive the time taken for the balls to collide:

Since both balls start from rest, we can use the equations of motion. The time taken for each ball to travel their respective distances can be expressed as:

• Time for ball M: t_M = √(2s_M/a_M)
• Time for ball m: t_m = √(2s_m/a_m)

Setting these two times equal gives us the relationship needed to find θ. After some algebra, we find that:

Part (b): Time Until Collision

Now that we have the angle θ, we can find the time it takes for the balls to collide after the string breaks. We know that the distance each ball travels is related to the angle and the radius:

Using the previously derived equations for distance and acceleration, we can substitute θ into the time equations:

For ball M, the time until collision can be expressed as:

Substituting θ from part (a) gives us:

Part (c): Second Collision Location

Assuming the first collision is perfectly elastic, we need to determine where the balls will collide again after the first collision. In a perfectly elastic collision, both momentum and kinetic energy are conserved. The velocities of the balls after the first collision can be calculated using the conservation laws:

• Final velocity of ball M: v_M' = (M - m)/(M + m) * v_M + (2m)/(M + m) * v_m
• Final velocity of ball m: v_m' = (2M)/(M + m) * v_M + (m - M)/(M + m) * v_m

After the first collision, the balls will continue to move along the circular track. The distance they travel before colliding again can be calculated using their new velocities and the time it takes for them to meet again. The angle through which each ball travels can be calculated based on their velocities and the time until the next collision.

Ultimately, the second collision will occur at an angle that can be expressed in terms of the initial conditions and the masses of the balls. The exact position can be derived from the equations of motion, taking into account the new velocities and the time calculated previously.

In summary, the collision angle, time until collision, and the location of the second collision can all be derived from the principles of conservation of momentum and energy, along with the equations of motion on a circular track.