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 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