Lessons in Physics

Introduction

In this lesson, we will cover kinetic and potential energy. We will learn how to calculate kinetic and potential energy and why they are important.

Check out our other posts in lessons in physics:
Understanding Forces in Physics
Newton’s Laws of Motion

Kinetic vs Potential Energy

Kinetic energy is the energy an object has because of its motion. Potential energy is the energy an object has because of its position relative to other objects.

Kinetic Energy (KE)

To find the kinetic energy of an object, multiply the mass times the velocity squared and divide by two. If you are using English units, mass is in lbs and velocity is in feet per second, which gives you foot-pounds (ft-lbf). If you are using SI units (metric), your mass is in kilograms, the velocity is in meters per second and the result is in Joules (kg*m²/s²).

Unlike a force, kinetic energy does account for velocity. In fact, it is the most influential portion of the equation. This is because the velocity variable in the equation is squared. This is significant because if you double the mass of an object, you get double the kinetic energy (assuming everything else stays the same). However, if you double the velocity of the object, you get four times the kinetic energy!

Why is this important? Let’s look at some examples.

The regulation weight of a baseball in Major League Baseball is between 5.00 and 5.25 ounces (0.3125 and 0.3281 lb). The fastest pitch ever thrown (recorded) was 105.8 mph or 155.17 feet per second. Plugging this into the formula for kinetic energy, we get:

Contrast that with a 9mm bullet, which weights 115 grains (0.01643 lb) on the light end, and travels approximately 1200 feet per second. Plugging this information in the kinetic energy formula we get:

As you can see, a 9mm bullet weighs just 5% of the weight of the baseball, yet has three times the kinetic energy. This is due to its velocity being so much higher and that velocity is squared.

Potential Energy (U)

Potential energy is dependent on the object’s location, mass, and gravity. In the potential energy equation, m is the mass, g is the gravitational acceleration, and h is the objects height above another object or the ground depending on the system.

Because mass and gravity are often constants in a system being analyzed, the defining factor of an object’s potential energy is its height above the ground or another object. You can also think of it as how much energy an object “potentially” would release if allowed.

Dams and water towers are two examples of potential energy. Water is either stored at an elevated level, or the mass of the body of water is increased, or both.

Kinetic and Potential Energy Relationship

Kinetic energy and potential energy are equal to each other in a closed system, assuming that energy is conserved. However, it should be noted that the maximum value of the kinetic energy only occurs at the minimum value of the potential energy and vice versa.

In the example above about the dams and water towers, when the water is released to go to the homes in the network, or to spin the turbine to make power, it is converting from potential energy to kinetic energy.

Take a roller coaster car sitting at the top of a hill. When the roller coaster car is at the highest point, just before it starts the drop, it has the highest potential energy and the least amount of kinetic energy. As the car starts down the hill, it is turning the potential energy into kinetic energy by trading height for velocity. If it were to go up a second hill that was equally as tall as the first, it would turn all its kinetic energy back into potential energy. Roller coasters usually make the first hill the tallest so the cars always maintain kinetic energy throughout the ride.

Let’s solve a problem.

Let’s look at a few problems related to the roller coaster. We will assume no losses to friction or air resistance and that energy is 100% conserved.
1. What is the potential energy of the roller coaster car at position 1?
2. What is the velocity of the car at position 3?
3. At position 2, what is a) the remaining potential energy? b) the velocity?

Problem 1. To solve this problem, you will need the potential energy equation. We know the mass of the roller coaster car (500 kg) and the height of 32 m. Since these are SI units, we must use 9.8 m/s² as the gravitational acceleration.

Problem 2. Now that we know what the potential energy of the roller coaster car is at the top of the hill, we can use that number to find the velocity of the car at the bottom of the hill.

Problem 3a. In order to find the potential energy remaining in the roller coaster car, subtract the second position from the first position.

Problem 3b. Once the remaining potential energy is known, find the energy converted so far (not what remains) to find the velocity. In this example, the car has converted half of its potential energy to kinetic energy so the amount of converted energy is the same as what remains.

Make it easy

Now that we see the long way to calculate these answers, let’s do a little algebra and get a quick way to make the calculations. When calculating the velocity gained from the change in potential energy, the equation looks like this:

On the left side of the equation we have two sets of instances of “mg”. We can factor those out.

We have a mass term (m) on both sides of the equals sign. That means that they cancel each other, so we can further simplify the equation.

Now let’s move the 1/2 over to the other side, by dividing both sides by 1/2. When you divide by a half, you are multiplying by 2.

Last, to find V we need to take the square root of both sides.

Another way this equation may be written is:

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