Analyzing Conservation of Energy Graphs

Instructor: Amanda Robb

Amanda holds a Masters in Science from Tufts Medical School in Cellular and Molecular Physiology. She has taught high school Biology and Physics for 8 years.

In this lesson, we'll be learning how to use graphs to apply the law of conservation of energy to different situations in physics, such as a spring, pendulum, or a roller coaster.

What is the Law of Conservation of Energy?

Have you ever been to an amusement park? Many of us love the thrill of zooming down from the top of a roller coaster through the park. But, how do roller coasters work? First, a chain drags the roller coaster to the top of the track. We're all familiar with the anxiety-provoking 'click-click-click' noise as we rise higher and higher. At the top, the roller coaster cart momentarily pauses and then you're racing down the track, zooming through loops and turns. But, if the roller coaster doesn't have a motor like a car, how do you move so quickly?

The answer lies in the law of conservation of energy. This law states that energy can never be created or destroyed, only converted between types. As the roller coaster cart gets dragged to the top of the track its potential energy increases. Potential energy is stored energy, and the roller coaster has a particular kind called gravitational potential energy, or stored energy due to height. When the cart crests the hill, gravity pulls the cart down the hill and that potential energy is converted to kinetic energy or energy due to movement.

The higher the track goes, the greater gravitational potential energy the roller coaster will have, and thus the more kinetic energy it will have as it rolls down the track. This is why taller roller coasters go faster.

Energy Graphs

To study energy conversions such as the roller coaster, scientists can use energy graphs, where the amount of each type of energy is graphed over time for a particular object. Let's look at how to set the graph up first.

Time is the independent variable, so it always goes on the x, or horizontal axis on a graph. The y axis is the dependent variable, or what we measure. In our case, it will be the amount of energy measured in Joules (J).

Since we are talking about energy conversions, there will always be more than one type of energy on the graph, so we're going to need a key. Most energy graphs will have potential and kinetic energy, but other types of energy might also be on the graph.

At any given point, an object on the graph will have the same amount of total energy. However, that total energy will be distributed in different amounts of kinetic versus potential energy. Let's look at a few examples.


Roller Coaster

Like we explained earlier, a roller coaster is a perfect example of energy conversions. Let's look at how to use an energy graph to analyze the movement of a roller coaster.

Energy graph for a cart traveling down a roller coaster track
roller coaster graph

Based on this graph, what is happening to the motion of the coaster? We can see that at the initial time, the cart has all potential energy and no kinetic energy. So, we can infer that the roller coaster must be at its maximum height, but not moving. This is when the roller coaster is at the top of the hill.

In the next 20 seconds, the roller coaster's potential energy decreases, so it must start decreasing in height. At the same time, the kinetic energy increases, because energy cannot be created or destroyed. So, the cart must also be speeding up. At 50 seconds, the cart has all kinetic energy and no potential energy. So, at this point, the roller coaster must be moving at its maximum speed and have no height. This is when the roller coaster speeds across the ground on a flat piece of track.

At any point in time, the sum of kinetic and potential energy is always equal to 500 J, since energy can never be created or destroyed. At time 0, all 500 J is in potential energy. But, over time, the gravitational potential energy decreases and the kinetic energy increases until, at 50 seconds, all 500 J is in kinetic energy.


Have you ever played with a slinky as a kid? A spring, like a slinky, is another excellent example of energy conversions. Springs have a type of potential energy called elastic potential energy, or stored energy in a spring. When you press down on a spring you compress the elastic coils. In this case, the spring has elastic potential energy. When you let the spring go, its elastic potential energy is converted into kinetic energy and the spring jumps away.

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