Conservation of Energy

The principle

Energy is never created or destroyed. It can only be transformed from one form to another.

The total energy of an isolated system (no external heat or work crossing the boundary) is constant.

E_total = constant

This is the first law of thermodynamics. Mathematically expressed:

ΔU = Q − W

where U is the internal energy, Q is heat added to the system, and W is work done by the system.

Forms of energy

Form	Formula	Example
Kinetic (linear)	½mv²	Moving cars, projectiles
Kinetic (rotational)	½Iω²	Spinning wheels, planets rotating
Gravitational PE (near Earth)	mgh	Books on shelves, water in dams
Gravitational PE (general)	−Gm₁m₂/r	Planets, satellites
Spring (elastic) PE	½kx²	Compressed/stretched springs
Thermal	~NkT (for ideal gas)	Heat in bulk objects
Chemical	Bond energies	Fuel, food, batteries
Electrical	½CV² (capacitor)	Electric circuits, lightning
Magnetic	½LI² (inductor)	Inductors, electromagnets
Nuclear	Binding energy	Fission, fusion
Mass-energy	mc²	Particle/antiparticle, mass defect
Photon (electromagnetic)	hf or pc	Light, X-rays, radio

Mechanical energy and conservation

Mechanical energy = KE + PE. Conserved when only conservative forces act (gravity, springs, electric).

For an object in gravity (no friction):

½mv₁² + mgh₁ = ½mv₂² + mgh₂

This solves countless problems — pendulums, roller coasters, projectiles, spring launchers — without integrating forces over time.

Pendulum example

A pendulum of length L released from angle θ. Find max speed at bottom.

At top — height h = L(1 − cos θ), v = 0. At bottom — h = 0, all energy is KE.

mg·L(1 − cos θ) = ½mv²
v_max = √(2gL(1 − cos θ))

Roller coaster

Total mechanical energy at any point on a frictionless coaster equals starting energy. Speed at any point depends only on height drop, not the path. A 50 m drop gives v = √(2·9.8·50) ≈ 31 m/s, regardless of loops or curves.

Dissipation — energy lost to heat

Real systems have non-conservative forces — friction, air drag, internal damping. These convert mechanical energy into thermal energy (heat).

The total energy is still conserved (it's just relocated):

(KE + PE)_initial = (KE + PE)_final + Q_dissipated

where Q_dissipated is the energy now in thermal form.

Examples:

• Car braking: KE → heat in brake pads.
• Falling object reaching terminal velocity: gravitational PE → air heat.
• Two balls colliding inelastically: KE → heat + sound + deformation.
• Spring losing oscillation amplitude: KE/PE → internal heat (damping).

Mass-energy equivalence

Einstein's famous equation:

E = mc²

Mass IS a form of energy. The conversion factor c² = (3×10⁸)² = 9×10¹⁶ J/kg makes a small mass equivalent to enormous energy.

Mass	Energy equivalent
1 g (typical paperclip)	9 × 10¹³ J ≈ 25 GWh
1 kg (1 liter water)	9 × 10¹⁶ J ≈ 21.5 megatons of TNT
Atomic mass unit (1 u)	931.5 MeV
Electron rest mass	0.511 MeV
Proton rest mass	938.3 MeV

Nuclear reactions exploit this — fusion reactors convert ~0.7% of mass to energy. Fission reactors ~0.1%. Annihilation (matter + antimatter) is 100%.

"Mass conservation" and "energy conservation" merge into one law in relativity. Mass is just frozen energy.

JavaScript — energy conservation in pendulum

// Simple pendulum: simulate with energy method (no friction)
function pendulumEnergy(theta_init, length = 1, g = 9.81) {
  const m = 1; // unit mass
  const PE_init = m * g * length * (1 - Math.cos(theta_init));
  // At bottom, all PE → KE: ½mv² = PE_init → v = √(2·g·L(1-cos θ))
  const v_max = Math.sqrt(2 * g * length * (1 - Math.cos(theta_init)));
  
  return {
    PE_at_top: PE_init,
    KE_at_bottom: PE_init,  // by conservation
    v_max
  };
}

console.log(pendulumEnergy(Math.PI / 4));  // 45° release, length 1m

// Falling object hits ground
function fallingObject(mass, height, dragForce = 0) {
  const PE = mass * 9.81 * height;
  // With drag: KE_final = PE - drag_force × height
  const KE = PE - dragForce * height;
  if (KE < 0) return { reached_ground: false };  // wouldn't reach ground
  const v = Math.sqrt(2 * KE / mass);
  return {
    PE_initial: PE,
    KE_final: KE,
    energy_lost_to_heat: dragForce * height,
    v_at_ground: v
  };
}

// 1 kg from 100 m, no drag
console.log(fallingObject(1, 100));
// 1 kg from 100 m, drag force 5 N (continuous)
console.log(fallingObject(1, 100, 5));

// Verify mechanical energy is conserved at every point of pendulum swing
function pendulumStep(theta, omega, dt, length = 1, g = 9.81) {
  // Simple pendulum equation: theta'' = -(g/L) sin theta
  const alpha = -(g / length) * Math.sin(theta);
  const newOmega = omega + alpha * dt;
  const newTheta = theta + newOmega * dt;
  return [newTheta, newOmega];
}

let theta = Math.PI / 4, omega = 0;
const m = 1, L = 1, g = 9.81, dt = 0.001;
for (let t = 0; t < 1; t += dt) {
  [theta, omega] = pendulumStep(theta, omega, dt, L, g);
}
const KE = 0.5 * m * (L * omega) ** 2;
const PE = m * g * L * (1 - Math.cos(theta));
console.log(`Total mechanical energy: ${(KE + PE).toFixed(4)} J (should be ~${(0.5 * (1 - Math.cos(Math.PI/4))).toFixed(4)})`);

Where energy conservation shows up

• Mechanics — fastest way to find speeds. Pendulums, roller coasters, projectile heights, springs, gravitational orbits.
• Thermodynamics. First law: ΔU = Q − W. Engine efficiency, refrigeration, heat transfer.
• Electromagnetism. Capacitor charging/discharging, inductor energy storage, electromagnetic radiation.
• Quantum mechanics. Energy eigenvalues of bound states, transitions emit photons of specific energies.
• Particle physics. Decay analysis, scattering experiments — total energy in = energy out (including rest mass).
• Cosmology. Conservation of mass-energy in expanding universe (subtle — depends on what "system" means in curved spacetime).
• Engineering. Power plants, electric grids, vehicle dynamics — all balance energy inputs and outputs.

Common mistakes

• Confusing mechanical with total energy. Mechanical (KE + PE) is conserved only with conservative forces. Total energy is conserved always (when including heat, etc.).
• Forgetting heat in dissipative situations. Friction "loses" mechanical energy, but it converts to heat — total energy is unchanged.
• Treating mass and energy separately at high speeds. Relativity unifies them. Particle creation/annihilation requires E = mc² accounting.
• Believing in perpetual motion. Cannot create energy from nothing. Cannot run forever without losses (second law).
• Wrong PE reference point. Gravitational PE = mgh requires choosing where h = 0. Different choices give different PE values, but ΔPE is the same.
• Sign errors with PE. Spring PE is +½kx² (positive when stretched OR compressed). Gravitational PE near Earth is +mgh (positive going up). Get the signs right.