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Chapter 5: Problem 119

Suppose an Olympic diver who weighs 52.0 kg executes a straight dive from a 10-m platform. At the apex of the dive, the diver is 10.8 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 m/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 5539.304 J. When the diver enters the water, their kinetic energy is also 5539.304 J and they enter the water at a speed of approximately 14.59 m/s. The diver does not do work on entering the water, as the work is done by the water on the diver to stop their motion.

Step by step solution

01

Calculate the potential energy of the diver at the apex of the dive

To calculate the potential energy (PE) of the diver, we can use the formula: PE = m x g x h where m is the mass of the diver (52.0 kg), g is the acceleration due to gravity (9.81 m/s²), and h is the height above the surface of the water (10.8 m). PE = 52.0 kg x 9.81 m/s² x 10.8 m Now we can calculate the potential energy of the diver.
02

Calculate the diver's potential energy

Using the formula from step 1, we find the potential energy of the diver: PE = 52.0 kg x 9.81 m/s² x 10.8 m = 5539.304 J So, the potential energy of the diver at the apex of the dive, relative to the surface of the water, is 5539.304 J.
03

Use conservation of mechanical energy to find the diver's kinetic energy at the surface of the water

As the diver falls and reaches the surface, their potential energy will be fully converted into kinetic energy (KE). We can use the principle of conservation of mechanical energy to determine the diver's kinetic energy, since there are no non-conservative forces acting on the diver like air resistance, during the fall. The conservation of mechanical energy is defined by: Initial Mechanical Energy = Final Mechanical Energy PE (initial) = PE (final) + KE (final) At the surface, PE (final) = 0, because the diver is at height h = 0. Therefore, PE (initial) = KE (final) 5539.304 J = KE (final) So, the kinetic energy of the diver when entering the water is 5539.304 J.
04

Calculate the speed of the diver when entering the water

Now, we will determine the speed of the diver when they enter the water. We know that the kinetic energy is equal to: KE = (1/2) * m * v² where m is the mass of the diver and v is the speed. From step 3, we know the kinetic energy of the diver at the surface of the water (5539.304 J). We can now calculate the speed (v) by rearranging the formula: v² = (2 * KE) / m Inserting the known values, we get: v² = (2 * 5539.304 J) / 52.0 kg Now we can calculate v.
05

Find the diver's speed

Using the formula from step 4, we can find the diver's speed: v² = (2 * 5539.304 J) / 52.0 kg v² ≈ 213.056 v ≈ √213.056 v ≈ 14.59 m/s So, the diver will enter the water at a speed of approximately 14.59 m/s.
06

Determine whether the diver does work on entering the water

When the diver enters the water, their speed slows down due to the resistance exerted by the water. In this scenario, the diver is not exerting any force on the water, and it’s the water that applies a force on the diver to stop their motion, thus, the work is done by the water on the diver, and not by the diver on the water. So, the diver does not do work on entering the water.

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