Problem 45 - University Physics with Modern Physics

Question

Bouncing Ball. A 650 -gram rubber ball is dropped from an initial height of \(250 \mathrm{m},\) and on each bounce it returns to 75\(\%\) of its previous height. (a) What is the initial mechanical energy of the ball, just after it is released from its initial height? (b) How much mechanical energy does the ball lose during its first bounce? What happens to this energy? (c) How much mechanical energy is lost during the second bounce?

Step-by-Step Solution

Answer

(a) 1592.5 J. (b) 398.125 J (lost as heat and sound). (c) 298.59375 J.

Step 1: Calculate initial gravitational potential energy

The initial mechanical energy of the ball is due to gravitational potential energy when it is released from the height of 250 m. We can calculate it using the formula for gravitational potential energy: \( E = mgh \), where \( m = 0.65 \) kg (mass of the ball), \( g = 9.8 \) m/s² (acceleration due to gravity), and \( h = 250 \) m (initial height). Substituting the values, we get: \( E = 0.65 \times 9.8 \times 250 = 1592.5 \) J.

Step 2: Calculate height after first bounce

To find how much energy is lost, we first calculate the height after the first bounce. The ball bounces back to \( 75\% \) of its previous height, so the new height is \( 0.75 \times 250 = 187.5 \) m.

Step 3: Calculate mechanical energy after first bounce

The mechanical energy after the first bounce is again gravitational potential energy at the new height: \( E' = mgh' \), where \( h' = 187.5 \) m. Thus, \( E' = 0.65 \times 9.8 \times 187.5 = 1194.375 \) J.

Step 4: Calculate energy lost during first bounce

The energy lost during the first bounce is the difference between the initial energy and the energy after the first bounce: \( \ \text{Energy lost} = 1592.5 - 1194.375 = 398.125 \) J. This energy is lost as heat and sound due to the deformation and friction during the bounce.

Step 5: Calculate height after second bounce

After the second bounce, the ball reaches \( 75\% \) of its height after the first bounce: \( 0.75 \times 187.5 = 140.625 \) m.

Step 6: Calculate mechanical energy after the second bounce

The gravitational potential energy after the second bounce is: \( E'' = mgh'' \), where \( h'' = 140.625 \) m. Thus, \( E'' = 0.65 \times 9.8 \times 140.625 = 895.78125 \) J.

Step 7: Calculate energy lost during second bounce

The energy lost during the second bounce is the difference between the energy after the first bounce and the energy after the second bounce: \( \ \text{Energy lost} = 1194.375 - 895.78125 = 298.59375 \) J. Again, this energy is lost as heat and sound.

Key Concepts

Gravitational Potential Energy

Gravitational potential energy is a type of energy that an object possesses due to its position in a gravitational field. Imagine a ball high up in the air—this ball has potential energy because if it were to fall, gravity would pull it to the ground. The higher the object, and the more massive it is, the more gravitational potential energy it holds.
In our exercise with the bouncing ball, this concept helps us determine the initial mechanical energy the ball has when it is dropped from a height. We use the formula for gravitational potential energy:

• \( E = mgh \)

The variables are:

• \(m\) for mass of the object (in this case, the ball).
• \(g\) for the acceleration due to gravity which is \(9.8 \, \text{m/s}^2\).
• \(h\) for the height from which it is released.

For the 650-gram rubber ball dropped from 250 meters, using this formula gives us its initial mechanical energy. It helps us understand how energy is stored due to elevation in the gravitational field.

Energy Conservation

Energy conservation is a principle stating that energy in a closed system remains constant—it can neither be created nor destroyed, only transformed from one form to another. In our bouncing ball scenario, when the ball falls, gravitational potential energy converts into kinetic energy. When the ball hits the ground, this energy is partly converted to sound and heat, and some is stored temporarily as elastic potential energy as the ball deforms. As the ball bounces back up, the process reverses, converting kinetic energy back to potential energy.
However, not all the energy converts back to potential energy. The ball only reaches 75% of its original height on rebound, showing us that some energy has been transformed into other forms, not stored as potential energy. This illustrates how a portion of energy is transferred to the environment in the form of heat and sound, which we observe as energy loss.

Energy Loss Calculations

Energy loss calculations help us determine how much energy is transformed into non-useful forms, like heat and sound, in physical activities. This is vitally important in understanding the dynamics of energy transformations and conservation.
During the first bounce of the ball, we calculate the energy before and after the bounce to find the energy lost. Initially, the mechanical energy is calculated as gravitational potential energy. After bouncing, the new height determines the new potential energy. The difference between these energy values shows the energy lost during the bounce:

• First bounce energy loss: \(398.125 \, \text{J}\)

Likewise, the energy lost during the second bounce is calculated after determining the lower height reached. Again, the energy before and after this bounce is calculated, showing the second bounce energy loss:

• Second bounce energy loss: \(298.59375 \, \text{J}\)

This process of calculating energy loss crucially informs us of the efficiency of mechanical systems and the various forms into which energy transforms.