Energy Graphs: Difference between revisions

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


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


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


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


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
3. Total Mechanical Energy
Total energy is:
Total energy is:
E = K + U
  E = K + U
 
For conservative systems, total energy is constant → horizontal line on graphs.
For conservative systems, total energy is constant → horizontal line on graphs.


Allowed Motion
Allowed Motion
Motion is only possible where:
Motion is only possible where:
E ≥ U(x)
  E ≥ U(x)




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




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At turning points:
At turning points:


  K = 0
  velocity = 0 (object reverses direction)


K = 0
velocity = 0 (object reverses direction)
[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:
Kinetic energy is:
K(x) = E – U(x)
  K(x) = E – U(x)
Since:
Since:
K = ½mv²
  K = ½mv²
 
 
High K → fast motion
 
 
Low K → slow motion


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




Important:
Important:


 
  K is always ≥ 0
K is always ≥ 0




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A. Spring Potential (Harmonic Oscillator)
A. Spring Potential (Harmonic Oscillator)
U(x) = ½kx²
  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]
  Parabola opening upward
  Minimum at x = 0 → stable equilibrium
  Motion is oscillatory


B. Gravitational Potential (Near Earth)
B. Gravitational Potential (Near Earth)
U = mgh
  U = mgh
 


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


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


  Negative potential energy
  Stronger interaction at small r


Negative potential energy
Stronger interaction at small r
[INSERT IMAGE: attractive potential curve approaching zero from below]


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


  Positive potential energy
  Objects are pushed apart


Objects are pushed apart
[INSERT IMAGE: repulsive potential curve approaching zero from above]


6. Bound vs Unbound Systems
6. Bound vs Unbound Systems
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E < 0
  E < 0
 
 
Object is trapped
 
 
Motion occurs between turning points


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


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


Unbound System
Unbound System


  E > 0


E > 0
  Object escapes
 
  Example: spacecraft leaving a planet
 
Object escapes
 
 
Example: spacecraft leaving a planet


[INSERT IMAGE: energy line above potential curve]


Escape Energy
Escape Energy
E = 0
  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
  Object barely escapes
<iframe src="https://trinket.io/glowscript/31d0f9ad9e" width="100%" height="600"></iframe>
  Final velocity approaches 0 at infinity


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


7. How to Read Any Energy Graph


read slopes


  Where U is low → speed is high


compare U and E
  Where U is high → speed is low


  U = E → turning point


find turning points


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


→ 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).
  K(x) = E – U(x) always

Revision as of 19:44, 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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