91Ó°ÊÓ

Potential energy is the stored energy an object has due to its position. In this context, when the 67.0-kg diver is standing atop a 3.00-meter-high tower, the object (the diver) holds gravitational potential energy. This energy depends on three factors:
\[ \text{Potential Energy} (PE) = mgh \]
where:

• \( m \) = mass in kilograms (67.0 kg)
• \( g \) = acceleration due to gravity (9.81 m/s²)
• \( h \) = height in meters (3.00 m)

The diver’s potential energy at the top of the tower is calculated by multiplying these values together, resulting in approximately 1971.9 Joules. This energy is what makes it possible for the diver to do work as they fall.
Understanding potential energy helps us see how energy is converted as the diver moves from a high point to rest in the water.

The work-energy principle is a critical concept in physics, explaining how work done on an object affects its energy state. Initially, the diver has gravitational potential energy at the top of the tower, which is transformed into kinetic energy as the diver falls. Once they hit the water, forces come into play to change this kinetic energy into work.
This principle states the total work done on an object is equal to its change in kinetic energy:
\[ W = \Delta KE \]
In this problem, the total mechanical energy is conserved until nonconservative forces like water resistance act. Thus, the work done by the water is equal to the change in mechanical energy from the top of the tower to rest underwater, approximately \(-1971.9 \text{ J}\) due to the negative sign indicating loss of energy.Understanding this principle is vital, as it gives insight into how forces and energy convert in practical scenarios, like the diver being slowed down by water.

Nonconservative forces are forces where the work done depends on the path taken. They do not have potential energies associated with them and often cause mechanical energy to dissipate, usually as thermal energy. Examples include friction and air resistance.
In the exercise, the water exerts a nonconservative force on the diver. As the dive enters into the water causing the person to decelerate over a distance of 1.10 meters, the kinetic energy is effectively dissipated.
Using the work-energy principle, we can calculate the average force exerted by the water as needed to halt the diver. Given that the work done \( W \) was \(-1971.9 \text{ J}\), and the displacement \( s \) in the water was 1.10 meters:
\[ F = \frac{W}{s} = \frac{-1971.9}{1.10} \approx -1792.6 \text{ N} \]
To find the magnitude, we take the absolute value, leading to the average force the water exerts on the diver being 1792.6 N. This illustrates how nonconservative forces can significantly influence motion by converting mechanical energy into other energy forms.