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| Energy Graphs in Physics
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| Energy graphs are one of the most powerful visualization tools in introductory physics.
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| They allow you to understand motion, stability, forces, and energy conservation without needing detailed algebra.
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| This page explains all major types of energy graphs used in Physics 1.
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| 1. What Is an Energy Graph?
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| An energy graph typically plots energy vs. position or energy vs. time.
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| Common types include:
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| Potential energy vs position, U(x)
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| Kinetic energy vs position, K(x)
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| Total energy vs position, E(x)
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| Energy vs time, E(t), K(t), U(t)
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| Energy graphs help you:
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| visualize where forces act
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| identify stable / unstable equilibrium
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| determine allowed motion
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| find turning points
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| understand speed without equations
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| 2. Potential Energy Graphs U(x)
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| Potential energy curves tell you everything about motion.
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| Force from U(x)
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| Force is the negative slope of U(x):
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| F = – dU/dx
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| if U slopes up, force points left
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| if U slopes down, force points right
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| Equilibrium Points
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| Equilibrium occurs where the slope = 0.
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| Minimum in U(x) → stable equilibrium
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| Maximum in U(x) → unstable equilibrium
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| 3. Total Mechanical Energy: E = K + U
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| Total energy E is constant for conservative systems.
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| Motion is allowed only where:
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| E ≥ U(x)
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| Turning points occur where:
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| E = U(x)
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| At those points, K = 0 → the object momentarily stops.
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| Insert image here:
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| 4. Kinetic Energy Graphs K(x)
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| Since K = ½mv²:
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| high K → fast motion
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| low K → slow motion
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| K = 0 → stopped
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| K is always ≥ 0
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| From a potential-energy graph:
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| K(x) = E – U(x)
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| This allows you to sketch velocity without solving equations.
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| 5. The Most Important Shapes to Know
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| A. Spring Potential Energy
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| U(x) = ½ k x² → a parabola opening upward
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| Key facts:
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| minimum at x = 0 → stable
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| total energy = horizontal line
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| K(x) = difference between E and U(x)
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| B. Gravitational Potential Energy (Near Earth)
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| U = mgh → linear in height
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| Great for sled/hill problems
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| Important insight:
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| Steeper does not mean faster. Only height difference determines final speed.
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| C. Attractive Gravitational/Electric Potentials
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| https://upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Electric_potential_energy_attractive.svg/640px-Electric_potential_energy_attractive.svg.png
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| This graph explains:
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| bound systems (E < 0)
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| escape energy (E = 0)
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| unbound states (E > 0)
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| D. Repulsive Electric Potentials
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| Positive potential energy that decreases as r increases.
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| https://upload.wikimedia.org/wikipedia/commons/thumb/4/45/Electric_potential_energy_repulsive.svg/640px-Electric_potential_energy_repulsive.svg.png
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| Used in proton–proton problems.
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| ---
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| 6. Bound vs Unbound Systems**
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| ### **Bound System**
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| https://upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Bound_state_potential.svg/640px-Bound_state_potential.svg.png
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| * total energy E < 0
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| * object cannot escape to infinity
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| * example: orbiting planet
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| ### **Unbound System**
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| https://upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Unbound_state_potential.svg/640px-Unbound_state_potential.svg.png
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| * total energy E > 0
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| * object can escape
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| * example: Voyager leaving the solar system
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| ### **Escape Speed Case**
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| https://upload.wikimedia.org/wikipedia/commons/thumb/1/19/Escape_velocity_energy.svg/640px-Escape_velocity_energy.svg.png
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| * E = 0 exactly
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| * object asymptotically approaches v → 0 as r → ∞
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| ---
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| 7. How to Read Any Energy Graph**
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| This is a checklist that helps on exams.
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| * Where U is **low**, speed is **high**
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| * Where U is **high**, speed is **low**
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| * Where U = E → turning point
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| * Slope of U → direction of force
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| * Minimum of U → stable equilibrium
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| * Maximum of U → unstable equilibrium
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| * K(x) = E – U(x) always
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| This allows you to solve conceptual problems quickly.
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| ---
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| 8. Example Problems (Exam Style)**
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| *Problem 1: Two Hills, Same Height**
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| Which is faster at the bottom?
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| **Same speed.**
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| Only **height** matters, not steepness.
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| ---
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| Problem 2: Object Sliding in a Potential Well**
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| Where is it fastest?
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| **Where U is minimum.**
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| ---
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| Problem 3: Proton and Electron Released**
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| Use attractive potential:
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| * U is negative
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| * object speeds up as U decreases
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| * motion allowed where K = E – U ≥ 0
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| ---
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| 9. Interactive Simulation (GlowScript/VPython)**
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| ```
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| <iframe src="https://trinket.io/glowscript/31d0f9ad9e" width="100%" height="600"></iframe>
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| ```
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| Students can:
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| * adjust potential functions
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| * launch particles
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| * visualize total energy, potential, kinetic
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| * see turning points and oscillations
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| ---
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