Question

A Super Ball has a coefficient of restitution of 0.8887 . If the ball is dropped from a height of \(3.853 \mathrm{~m}\) above the floor, what maximum height will it reach on its third bounce?

Short answer

Answer: The maximum height on the third bounce is approximately 2.705 meters.

Step-by-step solution

Initial height

The ball is initially dropped from a height of \(3.853 \mathrm{~m}\).

Determine the maximum height after the first bounce

The height reached after the first bounce can be determined by multiplying the initial height by the coefficient of restitution \(e\), so \(h_1 = 3.853 * 0.8887\e{latex} = h_1 = 3.853 * 0.8887 \approx 3.423 \mathrm{~m}\).

Determine the maximum height after the second bounce

The height reached after the second bounce is the height after the first bounce multiplied by the coefficient of restitution \(e\), so \(h_2 = 3.423 * 0.8887 \approx 3.043 \mathrm{~m}\).

Determine the maximum height after the third bounce

The height reached after the third bounce is the height after the second bounce multiplied by the coefficient of restitution \(e\), so \(h_3 = 3.043 * 0.8887 \approx 2.705 \mathrm{~m}\). The maximum height the Super Ball will reach on its third bounce is approximately \(2.705 \mathrm{~m}\).

Key concepts

Coefficient of Restitution

The coefficient of restitution (COR) is a crucial concept in physics, particularly when studying collisions and bouncing balls. This dimensionless quantity measures how elastic a collision is between two objects. An COR value of 1 indicates a perfectly elastic collision, meaning there's no loss in kinetic energy, while a value of 0 would represent a completely inelastic collision, where the objects do not separate after impact. In our exercise, a Super Ball with a COR of 0.8887 signifies that the ball retains 88.87% of its kinetic energy after bouncing off the floor.

For each bounce, the ball's maximum height post-collision can be predicted using the COR. If dropped from a height of 3.853 meters, the subsequent heights can be determined by successive multiplications of the previous height by the COR. This helps explain the step-by-step decrement in the ball's peak height after each bounce.

Elastic Collision

When two objects collide and rebound without any loss of kinetic energy, we call it an elastic collision. In reality, perfectly elastic collisions are rare, as some energy is usually transformed into other forms, like sound or thermal energy. However, the basis of elastic collisions in physics problems helps students understand energy transfer and conservation principles. The exercise involving the Super Ball is a practical demonstration of an almost elastic collision where the ball does not lose all its kinetic energy after striking the floor and continues to bounce.

In the context of our problem, the almost elastic properties of the ball, characterized by its coefficient of restitution, dictate how high the ball will bounce after each impact. Though it loses some energy, the collisions are sufficiently elastic to be modelled using this coefficient.

Conservation of Energy

The law of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. When applying this to physics problems, especially those involving kinematics and collisions, it provides a framework for understanding the mechanisms behind motion and energy exchange. In the case of the bouncing ball, the gravitational potential energy just before impact is converted predominantly into kinetic energy as the ball rises up after the bounce, and a small amount becomes heat or sound energy.

The coefficient of restitution outlines the efficiency of this energy transformation during each bounce. Calculating the maximum height reached on the third bounce requires the application of energy conservation principles, together with kinematic equations, to provide the step-by-step solution we see in the exercise.

Kinematics

Kinematics is the branch of mechanics that deals with motion without considering its causes. It involves the description of positions, velocities, and accelerations of objects. In our exercise, kinematics comes into play when determining how the motion of the Super Ball changes with each bounce. Kinematic equations describe the motion of an object under gravity's influence, which allows us to predict the ball's behavior after each collision with the floor.

The exercise simplifies kinematics by using the coefficient of restitution, which includes all the effects of the collision, to find the successive heights. This concept is essential in many physics problems as it helps describe an object's motion in a quantifiable and predictable manner, thus providing a solid basis for understanding more complex systems.