Elastic And Inelastic Collisions

We have two balls colliding head-on. We are given their masses, initial velocities, and the coefficient of restitution. We need to find their velocities after the collision.

1. Understanding the Relevant Equations:

Conservation of momentum: $ m_1 m_2 v_1 v_2 v_1' v_2' e m_1 = 1 \, \text{kg} v_1 = 12 \, \text{m/s} m_2 = 2 \, \text{kg} v_2 = -24 \, \text{m/s} e = \frac{2}{3} v_1' v_2' v_2' = -4 \, \text{m/s} v_1' = v_2' - 24 $

Final Answer: The correct answer is (B) -28 m/s, -4 m/s.

A ball hits the floor and rebounds after an inelastic collision. We need to determine what quantity is conserved in this scenario.

1. Understanding the Inelastic Collision:

Inelastic Collision: A collision where kinetic energy is not conserved.
Momentum Conservation: Momentum is conserved in a closed system.
Closed System: A system where no mass enters or leaves, and the net external force is zero.
Mechanical Energy: Sum of potential and kinetic energy.

2. Analyzing the Options:

• (A) The momentum of the ball is conserved: This is incorrect. During the collision, the floor exerts an external force on the ball. So, the momentum of the ball alone is not conserved. The ball's velocity (and hence momentum) changes direction and magnitude after the collision.
• (B) The mechanical energy of the ball is conserved: This is incorrect because the collision is specified as inelastic. In an inelastic collision, some of the kinetic energy is converted into other forms of energy like heat and sound.
• (C) The total momentum of the ball and the Earth is conserved: This is correct. The ball and the Earth can be considered a closed system. During the collision, the Earth exerts a force on the ball, and the ball exerts an equal and opposite force on the Earth (Newton's Third Law). Although the Earth's change in velocity is negligible due to its enormous mass, the momentum of the Earth changes. The total momentum of the ball and Earth remains constant.
• (D) The total mechanical energy of the ball and the Earth is conserved: This is incorrect. In an inelastic collision, the total kinetic energy of the ball and the Earth is converted into heat and sound energy. So, the total mechanical energy of the ball and the Earth is not conserved.

Final Answer:
The correct answer is (C) the total momentum of the ball and the Earth is conserved.