91Ó°ÊÓ

Kinetic energy is the energy of motion. It describes how much work an object can do based on its speed and mass. When the diver is about to hit the water, all of her initial potential energy, obtained from the height of the tower, is transformed into kinetic energy. The mathematical expression for kinetic energy is:\[ KE = \frac{1}{2}mv^2 \]where \( m \) represents mass and \( v \) is the velocity. In this exercise, it's essential to understand that just before the diver enters the water, her kinetic energy equals her initial potential energy—1971.39 J. Hence, no energy loss occurs, as air resistance is ignored. This energy conversion is critical for determining the work done by the water later on.

Potential energy is the stored energy of position possessed by an object. The diver's potential energy is based on her height above the water. We calculate it using:\[ PE = mgh \]where:

• \( m \) is the mass of the diver, 67.0 kg,
• \( g \) is the acceleration due to gravity, approximately \( 9.81 \text{ m/s}^2 \),
• \( h \) is the height of the jump, 3.00 m.

Given these values, the initial potential energy is 1971.39 J. As the diver falls, this energy is transformed seamlessly into kinetic energy. This exchange is what allows the diver to gain speed as she descends.

The work done by a force describes how much energy is transferred by the force acting over a distance. Here, when the diver hits the water surface, she begins to decelerate due to the force exerted by the water. The work done by the water, which brings her to a complete stop 1.10 m underwater, can be calculated as:\[ W = F \cdot d \]where:

• \( W \) is the work done by the water (-1971.39 J),
• \( F \) is the average force exerted by the water,
• \( d \) is the distance over which this force acts, 1.10 m.

The negative sign for work implies that the force exerted by the water opposes the motion, reducing the diver's kinetic energy.

The principle of conservation of energy states that energy in a closed system cannot be created or destroyed, only transformed from one form to another. In this scenario, the diver transitions from potential energy to kinetic energy smoothly without any energy loss to air resistance. When she enters the water, her kinetic energy is then spent entirely in doing work against the resisting force of the water. This transformation and balance of energy exemplify the conservation principle, as the total energy remains constant throughout her fall and submersion.

Calculating the force required to stop the diver involves understanding the relationship between work and force. From the work-energy principle, the force exerted by the water can be calculated by rearranging the work formula:\[ F = \frac{W}{d} \]Inserting the known values:

• Work \( W = -1971.39 \text{ J} \) (negative because the force direction is opposite),
• Distance \( d = 1.10 \text{ m} \),

we find the average force \( F \approx -1792.17 \text{ N} \). The magnitude of this force is 1792.17 N, which tells us how strongly the water acts to bring the diver to rest. This nonconservative force dissipates the diver's kinetic energy, that was otherwise conserved from the jump.