Problem 92 - Chemistry: The Central Science

Question

Suppose an Olympic diver who weighs \(52.0 \mathrm{~kg}\) executes a straight dive from a 10 -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. 5.93 The air bags that provide protection in automobiles in the event of an accident expand because of a rapid chemical reaction. From the viewpoint of the chemical reactants as the system, what do you expect for the signs of \(q\) and \(w\) in this process?

Step-by-Step Solution

Answer

(a) The potential energy at the apex of the dive is 5605.696 J. (b) The diver will enter the water at a speed of 14.7 m/s. (c) The diver does not do work on entering the water; instead, the water does work on the diver to bring the diver to a stop.

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

To calculate the potential energy at the apex of the dive, we can use the formula: PE = m*g*h where m is the mass of the diver (52 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 kg) * (9.81 m/s²) * (10.8 m) PE = 5605.696 J So the potential energy at the apex of the dive is 5605.696 J.

Step 2: Find the speed of the diver entering the water

According to the conservation of energy principle, the potential energy at the apex of the dive will be converted entirely into kinetic energy at the surface of the water. Therefore, we can write: PE = KE The formula for kinetic energy is: KE = (1/2) * m * v² where m is the mass of the diver (52 kg) and v is the velocity in m/s. Substituting the value of PE and m, we get: 5605.696 J = (1/2) * (52 kg) * v² Now, solve for v: v² = (5605.696 J * 2) / (52 kg) v² = 215.9872 v = sqrt(215.9872) = 14.7 m/s So, the diver will enter the water at a speed of 14.7 m/s.

Step 3: Determine if the diver does work on entering the water

When the diver enters the water, the water exerts an opposing force on the diver, which slows down the diver and decelerates them to a stop. According to the work-energy theorem, work is done on an object when an external force changes its kinetic energy. In this case, the external force is the water's opposing force, and it does work on the diver by reducing the diver's kinetic energy to zero. So, yes, the diver receives work as he/she enters the water. The diver does not do work on entering the water; instead, the water does work on the diver to bring the diver to a stop.

Key Concepts

Understanding Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. In simpler terms, anything that is moving has kinetic energy. The formula to calculate kinetic energy (\( KE \)) is:

\[ KE = \frac{1}{2} m v^2 \]

• \( m \) is the mass of the object.
• \( v \) is the velocity or speed of the object.

For example, when the Olympic diver makes their descent from the diving platform and gains speed, their potential energy converts into kinetic energy. By the time the diver reaches the water, all the potential energy has become kinetic energy, which we can calculate using the given formula.

The conversion of energy shows us a fundamental principle that when energy is transferred from one form to another, the total energy remains the same.

The Principle of Conservation of Energy

The conservation of energy is a vital concept in physics, declaring that energy cannot be created or destroyed. Instead, it changes from one form to another. This principle means that the total amount of energy in an isolated system remains constant.

In the context of the Olympic diver:

• At the top of their dive, the diver has maximum potential energy and no kinetic energy.
• As the diver falls, potential energy decreases while kinetic energy increases.
• Just before hitting the water, all the potential energy has converted into kinetic energy.

This amazing transformation adheres strictly to the conservation of energy principle, illustrating that the energy lost is exactly balanced by the energy gained. Thus, the dive and descent serve as a great demonstration of energy conservation in action.

Exploring the Work-Energy Theorem

The work-energy theorem states that work done by external forces on an object leads to a change in the object's kinetic energy.

When the Olympic diver hits the water, the water exerts a force that slows the diver down. Here’s how the work-energy theorem applies:

• The diver has kinetic energy upon entering the water.
• The water exerts an upward force, opposing the diver's motion.
• This force does work on the diver, turning kinetic energy into work as it brings the diver to a stop.

In summary, the theorem emphasizes the direct relationship between work and changes in kinetic energy. In this scenario, it's the water that does the work to absorb the diver’s energy, eventually halting their movement. Thus, understanding this theorem is crucial as it highlights how forces in the environment can influence the energy states of objects.