Final answer:
The motion involves energy conversion between kinetic and potential energy, with ball B reaching maximum potential energy at θmax and θmin, and maximum kinetic energy at θ = 0. Cart A's velocity will be adjusted according to the change in motion of ball B, based on the conservation of momentum.
Step-by-step explanation:
The question relates to the physics concept known as conservation of momentum and energy in the context of pendular motion and collisions within an isolated system. When cart A is given an initial velocity and ball B is at rest, the force that accelerates ball B will be the component of the tension in the string that acts horizontally, as there's no friction resistance on the track. During the motion, the total energy of ball B will be conserved, converting between potential energy when at its highest at θmax and θmin, and kinetic energy when passing through the lowest point at θ = 0. As the system consists of pendular motion, for a given displacement, the velocity of the pendulum and cart at various angles can be determined by using energy conservation.
-
- (a) At θ = θmax, ball B is momentarily at rest, as it has reached its maximum height and thus its velocity is zero. All its energy is potential.
-
- (b) At θ = 0, ball B passes through its lowest point, having maximum kinetic energy, and thus maximum velocity.
-
- (c) At θ = θmin, similar to the case at θmax, ball B is momentarily at rest at its minimum height (which should be symmetric to the maximum height if air resistance is negligible), with all its energy being potential.
Given that cart A and ball B have the same mass, the velocities can be tracked by considering the movement as a combination of the cart rolling and the pendulum swinging, respecting the conservation of angular momentum around the axis from which the ball B is suspended.