Energy Graphs

Claimed by Julianne Oliver

What is an Energy Graph?

Energy Diagrams are extremely useful for analyzing the interactions between two different objects. These diagrams can also be confusing if not properly understood. How do all of the different pieces of an energy diagram fit together? This page serves as a resource on how to construct and interpret energy diagrams. Let's get started!

The Basics

Energy diagrams are very useful for understanding different repulsive and attractive situations. These diagrams consist of three main components: kinetic energy, potential energy, and kinetic plus potential energy. These three energies are plotted as a function of energy versus separation between two objects.

Let's review:

Kinetic Energy: the energy that a moving object has in addition to its Rest Mass Energy. Kinetic energy is K on an energy graph.

Potential Energy: the energy associated with multiparticle systems. This is the energy between interacting objects, and this energy can be either attractive or repulsive. Potential energy is U on an energy graph.

What is the sum? Simply the sum of the potential and kinetic energies displayed on the graphs. We will discuss how to plot this value momentarily. K+U symbolizes the sum of the potential and kinetic energies.

Positive or Negative?

Kinetic energy can NEVER be negative! Potential energy is negative if the interaction is attractive and positive if the interaction is repulsive. The sum of the potential and kinetic energies depends on a few factors. If an object can "escape" from an attractive interaction, and, after having escaped, come to rest, then K+U is equal to zero. If however the object can escape with some final velocity, some final speed and motion, then K+U is greater than zero. Lastly, in a bound state, an object cannot escape the interaction it is involved in, and K+U is less than zero.

More Important Tidbits

K (kinetic energy) is zero or nonzero depending on the motion of the objects. At rest, kinetic energy is zero. In motion, kinetic energy is greater than zero. In an attractive interaction, such as Gravitational Potential Energy U (potential energy) increases (becomes less negative) and K decreases as two objects move farther away from one another. In a repulsive interaction, U(potential energy) decreases and K increases as the two objects move farther away from one another. The x-axis is separation, denoted r, and the y-axis is energy.

How to Construct an Energy Graph

Now that you know the basics, practicing constructing and analyzing energy graphs is the best way to learn! Here are some helpful steps for how to draw an energy graph:

1. Draw U first, based upon whether or not the interaction is attractive or repulsive.

2. Choose an r where you know k, and at this point plot the point (r, K).

3. Add the K value from step 2 to the U value at that same r value.

4. Plot K+U at this r value that you know both K and U.

5. Draw a horizontal line through this K+U point.

6. Find a K value at a different r value that, when added to U, gives K+U (the horizontal line you just drew).

7. With these two points on the K graph, sketch K vs. r.

A Computational Model

Vpython is great for modeling this concept. Using vpython, we can model many different systems that have kinetic and potential energy. We can model a spacecraft orbiting the Earth, and we can create graphs to display the kinetic, potential, and kinetic+potential energies of this system. See this code for how to do this!

[Sample Vpython code:https://trinket.io/glowscript/4010e21bc3]

Examples

See end for solutions to all examples.

Beginner

Label K, U, and K+U on the energy diagrams for the following situations:

Example 1: Two electrons are held at rest some finite distance apart, and they move away from each other after they are released. Their initial velocities are zero.

Example 2: A proton and an electron start out far apart. Their initial velocities are nonzero, and their energy is repulsive.

Example 3: An daughter has just enough energy to escape from her controlling mother (they have an attractive relationship).

Intermediate

Label K, U, and K+U on the following energy diagrams. THEN, label the following three energy graphs 1, 2, or 3 based on the following scenarios:

1: One of the components of this energy diagram is incorrect.

2: A proton and a electron are at rest, and they start out infinitely far apart.

3: An astronaut orbits the moon.

Advanced

Create an energy graph for each of the following situations.

Situation 1: A spacecraft is orbiting a moon. The spacecraft is given an initial velocity that allows the spacecraft to leave the moon's orbit with a final velocity greater than zero.

Situation 2: A boy jumps onto a merry-go-round and is attracted to the merry-go-round's axle. The boy's initial velocity is not high enough for him to escape the merry-go-round, so he continues to "orbit" the merry-go-round.

Situation 3: Two people are repulsed by on another and are trying to fight. They are held at rest by two of their friends a finite distant apart, and they of course move away from one another as soon as they are released, since their friends will not let them approach one another.

Solutions to Beginner Examples

Example 1:

Example 2: Example 3:

Solutions to Intermediate Examples

Example 1: K+U is incorrect! Example 2: Example 3:

Solutions to Advanced Examples

Situation 1: Situation 2: Situation 3:

Energy and My Future

How is energy related to my future? How is energy related to the things I am passionate about? How is energy and physics related to biochemistry, medical school, and surgery? In order to get into medical school, I must first take the MCATs. Physics and math majors score the highest on the MCATs, so physics is extremely important to master in order to successfully master the MCATs. Energy is everywhere, so I need to understand energy in order to be successful in the future. Physics is a part of everyday life. Energy is a part of everyday life. Mastering both now will only aid me in my dream of becoming a surgeon. From medical school student, to intern, to surgical resident, to, one day, a surgeon, energy and physics will follow me. Energy graphs only help to explain different energy interactions and make understanding energy simpler. Energy is a part of every scalpel I will hold, and kinetic energy will push me through late nights and long shifts. Is energy applicable to my future? Is energy applicable to medicine and surgery? Physics is more applicable to every aspect of my life than I ever could have imagined.

History

Gustave-Gaspard Coriolis first described "kinetic energy" in 1829, and William Rankine coined the term "potential energy" in 1853.

Energy was proven mathematically in 1918.

Emmy Noether (1882-1935) uncovered the fundamental justification for conservation laws.

Why is the discovery of energy so important? Energy and the conservation of energy drive our everyday lives. Everything we touch, eat, drink, use has energy! Without energy, there would be no life as we know it. The laptop you are using right now? Potential energy. Your hand scrolling through this page? Kinetic energy. Everything has energy! And these energy graphs are a great way to help us learn how to interpret energy in different situations.