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


* find turning points


* understand speed without equations
If U slopes up → force points left


2. Potential Energy Graphs U(x)


Potential energy curves tell you everything about motion.
If U slopes down → force points right


Force from U(x)


Force is the negative slope of U(x):
Steeper slope → stronger force


F = – dU/dx


if U slopes up, force points left


if U slopes down, force points right
[INSERT IMAGE: U(x) curve with slope arrows showing force direction]


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


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


Minimum in U(x) → stable equilibrium


Maximum in U(x) → unstable equilibrium
Maximum of U(x) → unstable equilibrium




3. Total Mechanical Energy: E = K + U


Total energy E is constant for conservative systems.
[INSERT IMAGE: potential well showing stable vs unstable equilibrium]


Motion is allowed only where:
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)
E ≥ U(x)


Turning points occur 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)
 


At those points, K = 0 → the object momentarily stops.


[INSERT IMAGE: horizontal energy line intersecting U curve at turning points]


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
High K → fast motion


low K → slow motion


K = 0 → stopped
Low K → slow motion
 
 
K = 0 → object stops
 
 
Important:
 


K is always ≥ 0
K is always ≥ 0


From a potential-energy graph:


K(x) = E – U(x)


This allows you to sketch velocity without solving equations.
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
 
 
 
[INSERT IMAGE: parabola with horizontal energy line and oscillation region]
 
B. Gravitational Potential (Near Earth)
U = mgh
 
 
Linear with height
 
 
Used for ramps and hills
 
 
Key idea:
Speed depends only on height difference, not slope.
 
[INSERT IMAGE: different slopes with same height drop]
 
C. Attractive Potentials (Gravity / Electric)
U(r) = –k/r
 
 
Negative potential energy
 
 
Stronger interaction at small r
 
 
 
[INSERT IMAGE: attractive potential curve approaching zero from below]
 
D. Repulsive Potentials
U(r) = +k/r
 
 
Positive potential energy
 
 
Objects are pushed apart
 
 
 
[INSERT IMAGE: repulsive potential curve approaching zero from above]
 
6. Bound vs Unbound Systems
 
Bound System
 
 
E < 0
 
 
Object is trapped
 
 
Motion occurs between turning points
 
 
Example: orbiting planet
 
[INSERT IMAGE: energy line below zero inside potential well]
 
Unbound System
 
 
E > 0
 
 
Object escapes
 


5. The Most Important Shapes to Know
Example: spacecraft leaving a planet


[INSERT IMAGE: energy line above potential curve]


A. Spring Potential Energy
Escape Energy
E = 0


U(x) = ½ k x² → a parabola opening upward


Object barely escapes


Key facts:


minimum at x = 0 → stable
Final velocity approaches 0 at infinity


total energy = horizontal line


K(x) = difference between E and U(x)


B. Gravitational Potential Energy (Near Earth)
[INSERT IMAGE: escape energy diagram]


U = mgh → linear in height
7. How to Read Any Energy Graph (Exam Checklist)
Great for sled/hill problems




Important insight:
Where U is low → speed is high
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
Where U is high → speed is low


This graph explains:


bound systems (E < 0)
U = E → turning point


escape energy (E = 0)


unbound states (E > 0)
Slope of U → direction of force


D. Repulsive Electric Potentials


Positive potential energy that decreases as r increases.
Steeper slope → stronger force


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.
Minimum → stable equilibrium
---


6. Bound vs Unbound Systems**


### **Bound System**
Maximum → unstable equilibrium


https://upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Bound_state_potential.svg/640px-Bound_state_potential.svg.png


* total energy E < 0
K(x) = E – U(x) always
* object cannot escape to infinity
* example: orbiting planet


### **Unbound System**


https://upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Unbound_state_potential.svg/640px-Unbound_state_potential.svg.png


* total energy E > 0
8. Common Mistakes
* object can escape
* example: Voyager leaving the solar system


### **Escape Speed Case**


https://upload.wikimedia.org/wikipedia/commons/thumb/1/19/Escape_velocity_energy.svg/640px-Escape_velocity_energy.svg.png
Thinking steeper hill means faster (wrong)


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


---
Letting kinetic energy be negative (impossible)


7. How to Read Any Energy Graph**


This is a checklist that helps on exams.
Ignoring forbidden regions


* Where U is **low**, speed is **high**
* 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.
Confusing force with value of U (it’s the slope, not the height)


---


8. Example Problems (Exam Style)**


*Problem 1: Two Hills, Same Height**
9. Example Problems


Problem 1: Two hills, same height
Which is faster at the bottom?
Which is faster at the bottom?
Answer: Same speed
Only height difference matters.


**Same speed.**
Problem 2: Where is the object fastest?
Only **height** matters, not steepness.
Answer: Where U is minimum.


---
Problem 3: Direction of force


Problem 2: Object Sliding in a Potential Well**


Where is it fastest?
Negative slope → force right


**Where U is minimum.**


---
Positive slope → force left


Problem 3: Proton and Electron Released**


Use attractive potential:


* U is negative
Problem 4: Where can the object move?
* object speeds up as U decreases
Answer: Only where E ≥ U(x)
* motion allowed where K = E – U ≥ 0


---
10. Advanced Insight
Energy graphs act like a “map of motion.”
From one graph, you can determine:


9. Interactive Simulation (GlowScript/VPython)**


speed (from kinetic energy)


```
 
acceleration (from slope)
 
 
direction (from slope sign)
 
 
This connects energy concepts directly to Newton’s Laws.
 
11. Interactive Simulation
<iframe src="https://trinket.io/glowscript/31d0f9ad9e" width="100%" height="600"></iframe>
<iframe src="https://trinket.io/glowscript/31d0f9ad9e" width="100%" height="600"></iframe>
```


Students can:
Final Takeaway
Energy graphs let you solve problems by reading instead of calculating.
If you can:
 
 
read slopes
 
 
compare U and E
 
 
find turning points
 


* adjust potential functions
→ you can solve most Physics 1 energy problems quickly.
* launch particles
* visualize total energy, potential, kinetic
* see turning points and oscillations


---
If you want next, I can make you a practice test that looks exactly like your exam (graphs + multiple choice traps).

Revision as of 19:38, 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 from Potential Energy 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


[INSERT IMAGE: U(x) curve with slope arrows showing force direction]

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


Minimum of U(x) → stable equilibrium


Maximum of U(x) → unstable equilibrium


[INSERT IMAGE: potential well showing stable vs 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)


[INSERT IMAGE: horizontal energy line intersecting U curve at turning points]

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


[INSERT IMAGE: parabola with horizontal energy line and oscillation region]

B. Gravitational Potential (Near Earth) U = mgh


Linear with height


Used for ramps and hills


Key idea: Speed depends only on height difference, not slope.

[INSERT IMAGE: different slopes with same height drop]

C. Attractive Potentials (Gravity / Electric) U(r) = –k/r


Negative potential energy


Stronger interaction at small r


[INSERT IMAGE: attractive potential curve approaching zero from below]

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


Positive potential energy


Objects are pushed apart


[INSERT IMAGE: repulsive potential curve approaching zero from above]

6. Bound vs Unbound Systems

Bound System


E < 0


Object is trapped


Motion occurs between turning points


Example: orbiting planet

[INSERT IMAGE: energy line below zero inside potential well]

Unbound System


E > 0


Object escapes


Example: spacecraft leaving a planet

[INSERT IMAGE: energy line above potential curve]

Escape Energy E = 0


Object barely escapes


Final velocity approaches 0 at infinity


[INSERT IMAGE: escape energy diagram]

7. How to Read Any Energy Graph (Exam Checklist)


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


8. Common Mistakes


Thinking steeper hill means faster (wrong)


Letting kinetic energy be negative (impossible)


Ignoring forbidden regions


Confusing force with value of U (it’s the slope, not the height)


9. Example Problems

Problem 1: Two hills, same height Which is faster at the bottom? Answer: Same speed Only height difference matters.

Problem 2: Where is the object fastest? Answer: Where U is minimum.

Problem 3: Direction of force


Negative slope → force right


Positive slope → force left


Problem 4: Where can the object move? Answer: Only where E ≥ U(x)

10. Advanced Insight Energy graphs act like a “map of motion.” From one graph, you can determine:


speed (from kinetic energy)


acceleration (from slope)


direction (from slope sign)


This connects energy concepts directly to Newton’s Laws.

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

Final Takeaway Energy graphs let you solve problems by reading instead of calculating. If you can:


read slopes


compare U and E


find turning points


→ you can solve most Physics 1 energy problems quickly.

If you want next, I can make you a practice test that looks exactly like your exam (graphs + multiple choice traps).

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