Introduction

When a ball is dropped from a height and strikes the ground, it undergoes a loss in kinetic energy. In this scenario, the ball loses 25% of its kinetic energy upon impact. We need to determine the height to which the ball bounces after hitting the ground.

Understanding the Loss in Kinetic Energy

When the ball is dropped from a height of 12m, it gains potential energy due to its position above the ground. As it falls, this potential energy is converted into kinetic energy. However, upon striking the ground, some of this kinetic energy is lost.

Accounting for the Loss in Kinetic Energy

The loss in kinetic energy can be accounted for by understanding the concept of energy conservation. According to the law of conservation of energy, the total energy of a system remains constant if no external forces are acting on it. In this case, the system consists of the ball falling from a height and bouncing back up.

When the ball strikes the ground, some of its kinetic energy is converted into other forms of energy, such as sound, heat, and deformation of the ball and the ground. Therefore, the total energy of the system is not conserved, and the ball loses a portion of its kinetic energy.

Determining the Height of the Bounce

To find the height to which the ball bounces, we need to consider the conservation of mechanical energy. The mechanical energy of the ball includes both its potential energy and kinetic energy.

Initially, when the ball is at rest at a height of 12m, it only possesses potential energy. As it falls, this potential energy is converted into kinetic energy. When the ball hits the ground, it loses 25% of its kinetic energy.

To find the height of the bounce, we can use the principle of energy conservation. The mechanical energy before the bounce is equal to the mechanical energy after the bounce, considering the loss in kinetic energy upon impact.

Let's assume the height to which the ball bounces is 'h'.

The mechanical energy before the bounce:
Potential energy = mgh (mass x acceleration due to gravity x height)
Kinetic energy = (1 - 0.25) x (1/2)mv^2 (where v is the velocity of the ball just before it hits the ground)

The mechanical energy after the bounce:
Potential energy = mgh (the height to which the ball bounces)
Kinetic energy = (1/2)mv^2 (where v is the velocity of the ball just after the bounce)

By equating the mechanical energies before and after the bounce, we can solve for 'h'.

Conclusion

In conclusion, when a ball is dropped from a height of 12m and loses 25% of its kinetic energy upon striking the ground, we can determine the height to which it bounces by applying the principle of energy conservation. By equating the mechanical energies before and after the bounce, we can solve for the height 'h'.

3 m