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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 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 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.