Energy Graphs

Energy Graphs in Physics - ALAYNA HASHMI

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?

• An energy graph typically plots energy vs. position or energy vs. time.

Common types include:

• Potential energy vs position, U(x)

• Kinetic energy vs position, K(x)

• Total energy vs position, E(x)

• Energy vs time, E(t), K(t), U(t)

• Energy graphs help you:

• visualize where forces act

• identify stable / unstable equilibrium

• determine allowed motion

• find turning points

• understand speed without equations

2. Potential Energy Graphs U(x)

Potential energy curves tell you everything about motion.

Force from U(x)

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 the slope = 0.

Minimum in U(x) → stable equilibrium

Maximum in U(x) → unstable equilibrium

3. Total Mechanical Energy: E = K + U

Total energy E is constant for conservative systems.

Motion is allowed only where:

E ≥ U(x)

Turning points occur where:

E = U(x)

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

4. Kinetic Energy Graphs K(x)

Since K = ½mv²:

high K → fast motion

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

Positive potential energy that decreases as r increases.

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**

1. 1. 1. **Bound System**

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

• total energy E < 0
• object cannot escape to infinity
• example: orbiting planet

1. 1. 1. **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
• object can escape
• example: Voyager leaving the solar system

1. 1. 1. **Escape Speed Case**

https://upload.wikimedia.org/wikipedia/commons/thumb/1/19/Escape_velocity_energy.svg/640px-Escape_velocity_energy.svg.png

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

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7. How to Read Any Energy Graph**

This is a checklist that helps on exams.

• 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.

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8. Example Problems (Exam Style)**

• Problem 1: Two Hills, Same Height**

Which is faster at the bottom?

• • Same speed.**

Only **height** matters, not steepness.

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Problem 2: Object Sliding in a Potential Well**

Where is it fastest?

• • Where U is minimum.**

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Problem 3: Proton and Electron Released**

Use attractive potential:

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

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9. Interactive Simulation (GlowScript/VPython)**

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

Students can:

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

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