A ball his the floor and rebounds after inelastic collision. In this case

Options

A. The momentum of the ball just after the collision is the same as that just before the collision.
B. the mechanical energy of the ball remains the same in the collision
C. the total momentum of the ball and the earth is conserved
D. the total energy of the ball and earth is conserved

Explanation

To solve the problem regarding the ball hitting the floor and rebounding after an inelastic collision, we can analyze the situation step by step.

### Step-by-Step Solution:

1. **Understanding the Collision**:
- When the ball hits the floor, it undergoes an inelastic collision. In an inelastic collision, the ball does not retain its initial velocity after rebounding. Instead, it bounces back with a reduced velocity.

2. **Defining Initial and Final Velocities**:
- Let the initial velocity of the ball just before the collision be \( u \).
- After the collision, the velocity of the ball when it rebounds is \( v \), where \( v < u \) because some kinetic energy is lost during the collision.

3. **Momentum Conservation**:
- If we consider the ball and the Earth as a system, the forces exerted by the Earth on the ball and the ball on the Earth are equal in magnitude and opposite in direction (Newton's Third Law).
- The net force on the system (ball + Earth) is zero, which implies that the total momentum of the system is conserved.
- Therefore, the momentum of the ball and Earth before and after the collision remains constant.

4. **Kinetic Energy Consideration**:
- In an inelastic collision, some kinetic energy is transformed into other forms of energy (like sound, heat, etc.). Thus, the final kinetic energy of the ball after the collision is less than the initial kinetic energy.
- This means that the mechanical energy of the ball does not remain the same during the collision.

5. **Total Energy of the System**:
- The total energy of the ball and Earth as a system is not conserved in the sense that some energy is dissipated to the surroundings. The energy lost is not accounted for in the ball-Earth system.

6. **Conclusion**:
- The correct statement regarding the situation is that the momentum of the ball and Earth as a system is conserved, while the mechanical energy of the ball is not conserved due to the inelastic nature of the collision.

### Final Answer:
The correct conclusion is that the momentum of the ball and Earth is conserved, while the mechanical energy of the ball is not conserved during the inelastic collision.