Problem 88 - Chemistry: The Central Science

Question

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.

Step-by-Step Solution

Answer

At the apex of the dive, the diver has a potential energy of 5526.696 J relative to the surface of the water. The diver enters the water at a speed of 14.58 m/s. As the diver enters the water, they exert a force on the water and cause a displacement, thus doing work on entering the water.

Step 1: Find the potential energy at the apex of the dive

To find the potential energy of the diver at the apex, we can use the potential energy formula: \[PE = mgh\] where \(PE\) is the potential energy, \(m\) is the mass of the diver, \(g\) is the acceleration due to gravity (approximately \(9.81 m/s^2\)) and \(h\) is the height. In this case, \(m = 52.0 kg\), \(g = 9.81 m/s^2\), and \(h = 10.8 m\). Plugging these values into the formula, we get: \[PE = (52.0 kg)(9.81 m/s^2)(10.8 m)\]

Step 2: Calculate the potential energy

Now, calculate the potential energy: \[PE = (52.0 kg)(9.81 m/s^2)(10.8 m) = 5526.696\, J\] So, the potential energy at the apex of the dive relative to the surface of the water is \(5526.696 J\).

Step 3: Apply conservation of mechanical energy

According to the conservation of mechanical energy, the potential energy at the apex of the dive will be converted entirely into kinetic energy at the surface of the water. So, we have: \[PE_{apex} = KE_{surface}\] Since the kinetic energy formula is: \[KE = \frac{1}{2}mv^2\] We can replace \(KE_{surface}\) by \(\frac{1}{2}mv^2\): \[PE_{apex} = \frac{1}{2}mv^2\] Now, substitute the potential energy we found in step 2: \[5526.696\, J = \frac{1}{2}(52.0 kg)v^2\]

Step 4: Calculate the speed of the diver at the surface

Solve the equation for the speed of the diver: \[v^2 = \frac{2(5526.696\, J)}{52.0 kg}\] \[v^2 = 212.56\] \[v = \sqrt{212.56} = 14.58 m/s\] So, the diver enters the water at a speed of \(14.58 m/s\).

Step 5: Determine if the diver does work on entering the water

When the diver enters the water, they are exerting a force on the water and causing a displacement. According to the definition of work, work is done when a force is exerted over a distance. In this case, as the diver enters the water, they are pushing the water away and out of their path, creating a displacement. Since there is both force and displacement involved, the diver does work on entering the water.