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Energy Graphs in Physics
THOMAS SCHIAVO FALL 2026
 
Energy graphs are one of the most powerful visualization tools in introductory physics.
They allow you to understand motion, stability, forces, and energy conservation without needing detailed algebra.
This page explains all major types of energy graphs used in Physics 1.


1. What Is an Energy Graph?
1. What Is an Energy Graph?
Energy graphs typically plot:


An energy graph typically plots energy vs. position or energy vs. time.
Common types include:


Potential energy vs position, U(x)
  Energy vs. position → U(x), K(x), E(x)


Kinetic energy vs position, K(x)


Total energy vs position, E(x)
  Energy vs. time → U(t), K(t), E(t)


Energy vs time, E(t), K(t), U(t)


Energy graphs help you:
They allow you to:
  visualize where forces act
  determine where motion is possible
  identify equilibrium points
  find turning points
  compare speeds instantly


visualize where forces act
identify stable / unstable equilibrium
determine allowed motion
find turning points
understand speed without equations


2. Potential Energy Graphs U(x)
2. Potential Energy Graphs U(x)
Potential energy graphs contain the most information.
  Force is the negative slope of the graph:
      F(x) = – dU/dx


Potential energy curves tell you everything about motion.
  If U slopes up → force points left
  If U slopes down → force points right
  Steeper slope → stronger force


Force from U(x)
[[File:Screenshot 2026-04-28 at 8.05.36 PM.png|center]]
 
Force is the negative slope of U(x):
 
F = – dU/dx
 
if U slopes up, force points left
 
if U slopes down, force points right


Equilibrium Points
Equilibrium occurs where:
  slope = 0 → F = 0


Equilibrium Points
Types:


Equilibrium occurs where the slope = 0.
  Minimum of U(x) → stable equilibrium
  Maximum of U(x) → unstable equilibrium


Minimum in U(x) → stable equilibrium


Maximum in U(x) → unstable equilibrium
3. Total Mechanical Energy
Total energy is:
  E = K + U


For conservative systems, total energy is constant → horizontal line on graphs.


3. Total Mechanical Energy: E = K + U
Allowed Motion
Motion is only possible where:
  E ≥ U(x)


Total energy E is constant for conservative systems.


Motion is allowed only where:
  If U > E → forbidden region
  If U = E → turning point


E ≥ U(x)


Turning points occur where:


E = U(x)
Turning Points
At turning points:


At those points, K = 0 → the object momentarily stops.
  K = 0
  velocity = 0 (object reverses direction)


Insert image here:
![K = E - U diagram](INSERT_YOUR_IMAGE_LINK)


4. Kinetic Energy Graphs K(x)
4. Kinetic Energy Graphs K(x)
Kinetic energy is:
  K(x) = E – U(x)
Since:
  K = ½mv²


Since K = ½mv²:
  High K → fast motion
 
  Low K → slow motion
high K → fast motion
  K = 0 → object stops
 
low K → slow motion
 
K = 0 → stopped
 
K is always ≥ 0
 
From a potential-energy graph:
 
K(x) = E – U(x)
 
This allows you to sketch velocity without solving equations.
 
5. The Most Important Shapes to Know
 
 
A. Spring Potential Energy
 
U(x) = ½ k x² → a parabola opening upward
 
 
Key facts:
 
minimum at x = 0 → stable
 
total energy = horizontal line
 
K(x) = difference between E and U(x)
 
B. Gravitational Potential Energy (Near Earth)
 
U = mgh → linear in height
Great for sled/hill problems
 
 
Important insight:
Steeper does not mean faster. Only height difference determines final speed.
 
C. Attractive Gravitational/Electric Potentials
 
https://upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Electric_potential_energy_attractive.svg/640px-Electric_potential_energy_attractive.svg.png
 
This graph explains:
 
bound systems (E < 0)
 
escape energy (E = 0)


unbound states (E > 0)


D. Repulsive Electric Potentials
Important:


Positive potential energy that decreases as r increases.
  K is always ≥ 0


https://upload.wikimedia.org/wikipedia/commons/thumb/4/45/Electric_potential_energy_repulsive.svg/640px-Electric_potential_energy_repulsive.svg.png


Used in proton–proton problems.
---


6. Bound vs Unbound Systems**
5. Most Important Potential Shapes


### **Bound System**
A. Spring Potential (Harmonic Oscillator)
  U(x) = ½kx²


https://upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Bound_state_potential.svg/640px-Bound_state_potential.svg.png
  Parabola opening upward
  Minimum at x = 0 → stable equilibrium
  Motion is oscillatory


* total energy E < 0
[[File:Screenshot 2026-04-28 at 8.01.36 PM.png|center]]
* object cannot escape to infinity
* example: orbiting planet


### **Unbound System**
B. Gravitational Potential (Near Earth)
  U = mgh


https://upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Unbound_state_potential.svg/640px-Unbound_state_potential.svg.png
  Linear with height
  Used for ramps and hills
  Speed depends only on height difference, not slope.


* total energy E > 0
[[File:Screenshot 2026-04-28 at 8.03.19 PM.png|center]]
* object can escape
* example: Voyager leaving the solar system


### **Escape Speed Case**
C. Attractive Potentials (Gravity / Electric)
  U(r) = –k/r


https://upload.wikimedia.org/wikipedia/commons/thumb/1/19/Escape_velocity_energy.svg/640px-Escape_velocity_energy.svg.png
  Negative potential energy
  Stronger interaction at small r


* E = 0 exactly
* object asymptotically approaches v → 0 as r → ∞


---
D. Repulsive Potentials
  U(r) = +k/r


7. How to Read Any Energy Graph**
  Positive potential energy
  Objects are pushed apart


This is a checklist that helps on exams.


* Where U is **low**, speed is **high**
6. Bound vs Unbound Systems
* Where U is **high**, speed is **low**
* Where U = E → turning point
* Slope of U → direction of force
* Minimum of U → stable equilibrium
* Maximum of U → unstable equilibrium
* K(x) = E – U(x) always


This allows you to solve conceptual problems quickly.
Bound System


---


8. Example Problems (Exam Style)**
  E < 0


*Problem 1: Two Hills, Same Height**
  Object is trapped
  Motion occurs between turning points
  Example: orbiting planet


Which is faster at the bottom?


**Same speed.**
Unbound System
Only **height** matters, not steepness.


---
  E > 0


Problem 2: Object Sliding in a Potential Well**
  Object escapes
  Example: spacecraft leaving a planet


Where is it fastest?


**Where U is minimum.**
Escape Energy
  E = 0


---
  Object barely escapes
  Final velocity approaches 0 at infinity


Problem 3: Proton and Electron Released**


Use attractive potential:
7. How to Read Any Energy Graph


* U is negative
* object speeds up as U decreases
* motion allowed where K = E – U ≥ 0


---
  Where U is low → speed is high


9. Interactive Simulation (GlowScript/VPython)**
  Where U is high → speed is low


  U = E → turning point


```
<iframe src="https://trinket.io/glowscript/31d0f9ad9e" width="100%" height="600"></iframe>
```


Students can:
  Slope of U → direction of force
  Steeper slope → stronger force
  Minimum → stable equilibrium
  Maximum → unstable equilibrium


* adjust potential functions
* launch particles
* visualize total energy, potential, kinetic
* see turning points and oscillations


---
  K(x) = E – U(x) always

Latest revision as of 20:06, 28 April 2026

THOMAS SCHIAVO FALL 2026

1. What Is an Energy Graph? Energy graphs typically plot: 


  Energy vs. position → U(x), K(x), E(x)



  Energy vs. time → U(t), K(t), E(t)


They allow you to: 

  visualize where forces act
  determine where motion is possible
  identify equilibrium points
  find turning points
  compare speeds instantly


2. Potential Energy Graphs U(x) Potential energy graphs contain the most information. 

  Force is the negative slope of the graph:
     F(x) = – dU/dx

  If U slopes up → force points left
  If U slopes down → force points right
  Steeper slope → stronger force

Equilibrium Points Equilibrium occurs where: 

  slope = 0 → F = 0

Types: 

  Minimum of U(x) → stable equilibrium
  Maximum of U(x) → unstable equilibrium


3. Total Mechanical Energy Total energy is: 

  E = K + U

For conservative systems, total energy is constant → horizontal line on graphs.

Allowed Motion Motion is only possible where: 

  E ≥ U(x)



  If U > E → forbidden region
  If U = E → turning point


Turning Points At turning points: 

  K = 0
  velocity = 0 (object reverses direction)


4. Kinetic Energy Graphs K(x) Kinetic energy is: 

  K(x) = E – U(x)

Since: 

  K = ½mv²

  High K → fast motion
  Low K → slow motion
  K = 0 → object stops


Important: 

  K is always ≥ 0


5. Most Important Potential Shapes

A. Spring Potential (Harmonic Oscillator) 

  U(x) = ½kx²

  Parabola opening upward
  Minimum at x = 0 → stable equilibrium
  Motion is oscillatory

B. Gravitational Potential (Near Earth) 

  U = mgh

  Linear with height
  Used for ramps and hills
  Speed depends only on height difference, not slope.

C. Attractive Potentials (Gravity / Electric) 

  U(r) = –k/r

  Negative potential energy
  Stronger interaction at small r


D. Repulsive Potentials 

  U(r) = +k/r

  Positive potential energy
  Objects are pushed apart


6. Bound vs Unbound Systems

Bound System 


  E < 0

  Object is trapped
  Motion occurs between turning points
  Example: orbiting planet


Unbound System 

  E > 0

  Object escapes
  Example: spacecraft leaving a planet


Escape Energy 

  E = 0

  Object barely escapes
  Final velocity approaches 0 at infinity


7. How to Read Any Energy Graph 


  Where U is low → speed is high

  Where U is high → speed is low

  U = E → turning point



  Slope of U → direction of force
  Steeper slope → stronger force
  Minimum → stable equilibrium
  Maximum → unstable equilibrium



  K(x) = E – U(x) always
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