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Potential energy is the energy that an object possesses due to its position relative to a reference point, in this case, the ground. It is an important concept when studying mechanical energy because it can be converted into kinetic energy when the object moves. This conversion happens when a stored potential energy is released.
When the basketball is held at a height of 1.05 meters, it has potential energy because of its position in Earth's gravitational field. The potential energy at this height can be calculated using the formula:

• \( E_p = mgh \), where \( m \) is the mass of the object, \( g \) is the acceleration due to gravity (approximately 9.81 m/s\(^2\)), and \( h \) is the height.

Using this formula, the potential energy of the basketball at 1.05 meters is calculated as 6.1893 Joules.
It's crucial to recognize that potential energy decreases when an object loses height and vice versa. By understanding potential energy, you can predict how an object like a basketball would behave when dropped and how high it might rebound.

The principle of energy conservation states that energy in a closed system cannot be created or destroyed; it can only be transformed or transferred from one form to another. This means the total mechanical energy (sum of potential and kinetic energy) before and after an event should remain constant if no external work is done.
In the case of the basketball, initially, all the mechanical energy is potential energy at 1.05 meters, and when it hits the ground, this potential energy converts into kinetic energy before rebounding to a lower height of 0.57 meters. The loss of mechanical energy can primarily be attributed to factors such as air resistance and energy dissipation when the ball strikes the ground — causing sound and heat production.
By applying the conservation of energy principle, we find the initial mechanical energy is greater than the rebounded energy (3.3522 Joules), indicating a loss of 2.8371 Joules during the collision, which is accounted for by the difference in heights.

Calculating the force exerted by the player in dribbling the ball involves understanding the work-energy principle, which states that work done on an object is equivalent to the change in its energy. In this context, the player does work on the basketball to restore its height to 1.05 meters, compensating for the energy lost during the bounce.
The work done, \( W \), is calculated as the product of the force exerted by the player and the distance over which this force acts:

• \( W = F \times d \)
• The lost energy, 2.8371 Joules, serves as the requirement for the player to bring the ball back to its original height.

Therefore, by rearranging the formula, the force can be found as follows:

• \( F = \frac{W}{d} \)
• This gives \( F = \frac{2.8371}{0.080} = 35.464 \) Newtons.

Understanding this calculation helps in grasping how external forces influence an object's energy state and allows students to connect concepts of energy, work, and force in practical scenarios.