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.

Short answer

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

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: \(PE = (52.0 \mathrm{kg})(9.81 \mathrm{m/s^2})(10.8 \mathrm{m})\)

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).

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: \(5564.976 \mathrm{J} = KE\)

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.

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.

Key concepts

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. When an object is moving, it has the ability to cause changes because of the energy it carries. For our Olympic diver, as they leave the platform and dive towards the water, their potential energy is transformed into kinetic energy.

To calculate kinetic energy (\( KE \)), use the formula:

• \( KE = \frac{1}{2}mv^2 \)

where \( m \) is the mass of the object (in kg) and \( v \) is its velocity (in m/s). As the diver descends, gravity accelerates them, increasing their speed and, consequently, their kinetic energy. At the point of entering the water, all potential energy is assumed to convert to kinetic energy, allowing us to determine the diver's speed upon impact.

Understanding kinetic energy helps in predicting how fast an object will move given its mass and energy. It also helps us see how energy changes form, especially important in scenarios where safety and precise movement, like professional diving, matter.

Conservation of Energy

The conservation of energy is a fundamental principle in physics stating that energy in an isolated system remains constant. It cannot be created or destroyed, only transformed from one form to another. For the diver in question, the potential energy they have at the apex of their jump transforms into kinetic energy as they fall.

Think of energy as currency that can be exchanged between different forms but never lost. Initially, the diver's potential energy is high because they are elevated. As they plunge towards the water, this energy gradually shifts to kinetic energy. What this exercise beautifully illustrates is that the total energy (potential + kinetic) remains constant during the dive. So, at the top of the leap, we have maximum potential energy, and just before hitting the water, we have maximum kinetic energy.

Understanding conservation of energy not only allows us to predict speeds and forces but also helps in confirming that energy has been accounted for, thereby ensuring no errors in calculations of the diver's descent.

Physics Problem Solving

Approaching physics problems systematically is key to mastering the subject. This particular problem serves as a great example of applying physics principles to solve real-world scenarios. Start by identifying the given values and what needs to be found. Here, we want to calculate potential and kinetic energy and understand energy transformation.

Next, write down the relevant formulas. Use the potential energy formula \( PE = mgh \) to find how much energy the diver has at peak height. Then, apply the principle of energy conservation to remember that this potential energy will turn into kinetic energy of the same amount just before the diver impacts the water. Hence, \( KE = PE \).

Finally, solve for the unknowns. In computing the diver's speed at water impact, equate kinetic energy \( KE = \frac{1}{2}mv^2 \) to the calculated potential energy. Solve to find the velocity, gaining insight on how fast the diver will be moving.

• Double-check each step for errors.
• Ensure consistency in units for correctness.

Practicing these problem-solving strategies fosters a deeper understanding and prepares one better for varied physics challenges.