To tackle this problem, we need to analyze the dynamics of the collision between the moving ball of mass M and the two identical balls connected by a spring. We will break this down into two parts: first, determining the minimum mass M required for the incident ball to strike the system again, and second, calculating the time elapsed between the two collisions.

Understanding the Collision Dynamics

In an elastic collision, both momentum and kinetic energy are conserved. When the ball of mass M collides with the first ball of mass m, it will transfer some of its momentum and energy to the two balls connected by the spring. The key here is to find the conditions under which the ball of mass M will collide with the system of two balls again after the initial impact.

Part (a): Minimum Mass M for Subsequent Collision

For the incident ball to strike the system again, it must have enough velocity after the first collision to travel back and collide with the two balls. Let's denote the initial velocity of mass M as V₀. After the collision, the first ball (mass m) will move with a certain velocity, and the second ball (also mass m) will remain at rest initially.

Using conservation of momentum, we can express the velocities after the collision:

• Initial momentum: MV₀
• Final momentum: MV' + mv₁ + mv₂

Where V' is the velocity of mass M after the collision, and v₁ and v₂ are the velocities of the two balls after the collision. Since the collision is elastic, we also have conservation of kinetic energy:

• Initial kinetic energy: (1/2)MV₀²
• Final kinetic energy: (1/2)MV'² + (1/2)mv₁² + (1/2)mv₂²

For the minimum mass M, we need to ensure that the velocity of the first ball (v₁) after the collision is sufficient to compress the spring and push the second ball (v₂) forward, allowing mass M to return and collide again. The condition for this is that the velocity of the first ball must be greater than zero after the collision.

By solving the equations of momentum and kinetic energy, we can derive that:

M ≥ 3m

Substituting m = 2 kg, we find:

M ≥ 6 kg

Part (b): Time Elapsed Between Collisions

To find the time between the two collisions, we need to analyze the motion of the two balls connected by the spring after the first collision. The system will oscillate due to the spring's restoring force. The time period T of a mass-spring system is given by:

T = 2π√(m/k)

For our system, we have two masses (each of mass m) connected by a spring with spring constant k. The effective mass for the oscillation can be considered as m/2 for each mass, leading to:

T = 2π√(2 kg / 1 kg/s²) = 2π√(2) seconds

Since the first collision will occur when the first ball moves back to its original position, the time elapsed between the two collisions will be half of the time period:

Time elapsed = T/2 = π√(2) seconds

Summary of Results

To summarize:

• The minimum mass M required for the incident ball to strike the system of two balls again is 6 kg.
• The time elapsed between the two collisions is π√(2) seconds.

This analysis illustrates the interplay between momentum, energy conservation, and harmonic motion in a system involving collisions and springs. If you have any further questions or need clarification on any part, feel free to ask!