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Chapter 5: Problem 92

Suppose an Olympic diver who weighs \(52.0 \mathrm{~kg}\) executes a straight dive from a \(10-\mathrm{m}\) platform. At the apex of the dive, the diver is \(10.8 \mathrm{~m}\) above the surface of the water. (a) What is the potential energy of the diver at the apex of the dive, relative to the surface of the water? (b) Assuming that all the potential energy of the diver is converted into kinetic energy at the surface of the water, at what speed in \(\mathrm{m} / \mathrm{s}\) will the diver enter the water? (c) Does the diver do work on entering the water? Explain.

Short Answer

Expert verified
The potential energy of the diver at the apex of the dive is approximately 5565 J. The diver enters the water with a speed of approximately 14.35 m/s. The diver does work when entering the water as they exert a force on the water, transferring kinetic energy into other forms of energy like heat and sound.

Step by step solution

01

Calculate the potential energy at the apex of the dive.

To calculate the potential energy (PE) of the diver at the apex of the dive, we use the formula for gravitational potential energy: \(PE = mgh\) where m is the mass of the diver (in kg), g is the acceleration due to gravity (approximated as \(9.81 \mathrm{m/s^2}\)), and h is the height above the surface of the water (in m). We are given m = 52.0 kg, and h = 10.8 m. Plugging these values into the formula, we get: \(PE = (52.0 \mathrm{kg})(9.81 \mathrm{m/s^2})(10.8 \mathrm{m})\)
02

Calculate the total potential energy at the apex of the dive.

Find the resulting potential energy by solving: \(PE = 5564.976 \mathrm{J}\) The potential energy of the diver at the apex of the dive, relative to the surface of the water, is approximately 5565 J (joules).
03

Apply the conservation of mechanical energy.

By using the conservation of mechanical energy, we can relate the potential energy to the kinetic energy (KE) at the surface of the water. The mechanical energy of the diver at the apex (potential energy) is equal to the mechanical energy at the surface (kinetic energy), so: \(PE = KE\) Since we know the potential energy at the apex, we can now solve for the kinetic energy at the surface by setting them equal: \(5564.976 \mathrm{J} = KE\)
04

Find the speed at which the diver enters the water.

Now we can use the equation for kinetic energy to solve for the diver's speed: \(KE = \frac{1}{2}mv^2\) where m is the mass of the diver and v is the speed (in m/s). We are given m = 52.0 kg, and we know the kinetic energy is equal to 5564.976 J. We need to solve for v: \(5564.976 \mathrm{J} = \frac{1}{2}(52.0 \mathrm{kg})v^2\) After solving for v, we get: \(v \approx 14.35 \mathrm{m/s}\) The diver will enter the water with a speed of approximately 14.35 m/s.
05

Determine if the diver does work on entering the water and explain.

When the diver enters the water, they exert a force on the water to slow down and come to a stop. Since work is defined as the force applied over a distance, it is clear that the diver does work when entering the water. This work is performed by the water's resistance against the diver's motion, which in turn decreases the diver's kinetic energy and transfers it into other forms of energy, such as heat and sound.

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