Question

A ball of mass m collides horizontally with a stationary wedge on a rough horizontal surface, in the two orientations as shown. Neglect friction between ball and wedge. Two student comment on system of ball and wedge in these situations

Saurav: Momentum of system in x-direction will change by significant amount in both cases. Rahul: There are no impulsive external forces in y-direction in both cases hence the total momentum of system in y-direction can be treated as conserved in both cases.

• A. Saurav is wrong and Rahul is correct
• B. Saurav is correct and Rahul is wrong
• C. Both are correct
• D. Both are wrong

Answer 1

Answer: (B)

Saurav is correct and Rahul is wrong

Answer 2

Analysis of Saurav's statement:

Saurav states: "Momentum of system in x-direction will change by significant amount in both cases."

The system consists of the ball and the wedge. The horizontal surface is rough. This means there is a friction force between the wedge and the ground. This friction force is an external force to the ball-wedge system.

In both Case I and Case II, the ball collides horizontally with the wedge, exerting a force on the wedge that has an x-component. This force will tend to move the wedge horizontally. Due to the rough surface, a friction force will act on the wedge from the ground, opposing this tendency of motion or actual motion.

During a collision, the forces involved are impulsive (large force acting for a short duration). The friction force from the ground, being an external force in the x-direction, will exert an impulse on the system. Since the surface is rough, this friction force can be significant, leading to a significant change in the x-momentum of the system. Therefore, the x-momentum of the system is not conserved. Saurav's statement is correct.

Analysis of Rahul's statement:

Rahul states: "There are no impulsive external forces in y-direction in both cases hence the total momentum of system in y-direction can be treated as conserved in both cases."

The external forces in the y-direction acting on the system are gravity (on the ball and wedge) and the normal force from the ground on the wedge. Gravity is a non-impulsive force. We need to check if the normal force from the ground can be impulsive.

• Case I (Ball hits inclined surface):

  When the ball hits the inclined surface, the normal force of interaction between the ball and the wedge (which is perpendicular to the inclined surface) has both a horizontal (x) and a vertical (y) component. The y-component of this normal force acts downwards on the wedge. Since the collision force is impulsive, this downward y-component is also an impulsive force. To prevent the wedge from moving vertically downwards (or to support it against this impulsive force), the normal force from the ground on the wedge must increase impulsively. Thus, the normal force from the ground is an impulsive external force in the y-direction. Since there is an impulsive external force in the y-direction, the total momentum of the system in the y-direction is not conserved.

• Case II (Ball hits vertical surface):

  When the ball hits the vertical surface, the normal force of interaction between the ball and the wedge is purely horizontal (in the x-direction). There is no vertical component of the collision force acting on the wedge. Therefore, the normal force from the ground on the wedge only needs to support the weight of the wedge (and the ball, if it were resting on it, which it is not during a horizontal collision). This normal force from the ground is not impulsive. Since there are no impulsive external forces in the y-direction, the total momentum of the system in the y-direction can be treated as conserved (it remains zero, as the ball has no initial y-momentum and no vertical forces are generated by the collision).

Since Rahul claims that momentum in the y-direction is conserved in both cases, and it is not conserved in Case I, Rahul's statement is incorrect.

Conclusion:

Saurav is correct, and Rahul is wrong.