Energy Graphs

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).