Energy and Work: Complete Physics Guide
Energy and work are two of the most fundamental concepts in physics. Work is done when a force causes displacement — it is the mechanism by which energy is transferred. Energy is the capacity to do work — a quantity that a system possesses and can transfer to other systems through work or heat.
The conservation of energy — that energy cannot be created or destroyed, only converted between forms — is one of the most powerful and universal principles in all of science. Understanding how kinetic energy (energy of motion), potential energy (stored energy), and work relate to each other gives you a powerful tool for solving problems without needing to track time or forces in detail.
Both work and energy are measured in Joules (J): 1 J = 1 N·m = 1 kg·m²/s². Other energy units: calorie (1 cal = 4.184 J), kilowatt-hour (1 kWh = 3.6 MJ), electron-volt (1 eV = 1.602×10⁻¹⁹ J).
Formula
W = F·d·cos(θ) | KE = ½mv² | PE = mgh
What Is Work?
Work is done by a force when the force has a component in the direction of displacement. W = F·d·cos(θ), where θ is the angle between the force vector and displacement vector. When force and displacement are parallel (θ = 0°), cos(θ) = 1 and W = Fd. When perpendicular (θ = 90°), cos(90°) = 0 and W = 0 — no work is done.
This is why carrying a heavy shopping bag horizontally at constant speed requires no mechanical work against gravity — the gravitational force is vertical but displacement is horizontal (θ = 90°). However, lifting the bag upward requires work: force and displacement are both upward (θ = 0°).
Negative work occurs when the force opposes displacement (θ = 180°): friction does negative work on a sliding object, removing kinetic energy. A braking force does negative work on a car, reducing its KE. Work can be positive, negative, or zero depending on direction.
Kinetic Energy — Energy of Motion
Kinetic energy is the energy an object possesses due to its motion: KE = ½mv². It depends on mass linearly but on velocity quadratically — doubling speed quadruples KE. A car at 60 km/h has 4× the KE of the same car at 30 km/h, which is why high-speed crashes are so much more destructive.
KE is always non-negative (v² ≥ 0, m > 0). An object at rest has zero KE. KE changes only when net work is done on the object — this is the Work-Energy Theorem: W_net = ΔKE = ½mv_f² − ½mv_i².
Examples: A 0.145 kg baseball at 40 m/s (90 mph) has KE = ½×0.145×1600 = 116 J. A 1500 kg car at 30 m/s (108 km/h) has KE = ½×1500×900 = 675,000 J = 675 kJ. The car has nearly 5800× more KE despite being only ~10,000× heavier, because v² scaling dominates.
| Energy type | Formula | Unit | Example |
|---|---|---|---|
| Kinetic Energy | KE = ½mv² | Joules (J) | Moving car, flying ball, running athlete |
| Gravitational PE | PE = mgh | Joules (J) | Raised weight, water behind dam, raised roller coaster |
| Elastic PE | PE = ½kx² | Joules (J) | Compressed spring, drawn bow, stretched rubber band |
| Work Done | W = F·d·cos(θ) | Joules (J) | Pushing a box, lifting a weight, braking a car |
| Power | P = W/t = Fv | Watts (W = J/s) | Engine output, human running, electrical appliance |
Potential Energy — Stored Energy
Gravitational potential energy is the energy stored in an object due to its height: PE = mgh, where h is measured from any chosen reference level (usually the lowest point in the problem). PE can be zero, positive, or negative depending on the reference — only changes in PE matter, not the absolute value.
Elastic potential energy is stored in deformed springs and elastic materials: PE = ½kx², where k is the spring constant (N/m) and x is the deformation from the natural length. A spring with k = 200 N/m compressed by 0.05 m stores PE = ½×200×0.0025 = 0.25 J.
Other forms of potential energy include chemical (batteries, food, fuel), nuclear (atomic nuclei), and electrostatic (charged objects). In all cases, PE represents stored energy that can be converted to kinetic energy when released.
Conservation of Mechanical Energy
Law of conservation of energy: Total energy of an isolated system remains constant. Energy can change form but cannot be created or destroyed. For mechanical systems with only conservative forces (gravity, spring): E = KE + PE = constant.
At the top of a roller coaster: KE = 0 (momentarily stationary or minimum), PE = maximum. At the bottom: PE = 0 (reference level), KE = maximum. The sum KE + PE is constant throughout, ignoring friction and air resistance.
The Work-Energy Theorem connects work by non-conservative forces (friction, applied forces) to changes in total mechanical energy: W_non-conservative = ΔKE + ΔPE = ΔE_mechanical. Friction always does negative work, reducing total mechanical energy and converting it to thermal energy.
Worked example: A 2 kg ball rolls down a frictionless ramp from h = 3 m. At the bottom (h = 0): PE lost = 2×9.81×3 = 58.86 J = KE gained. Speed v = √(2×KE/m) = √(2×58.86/2) = √58.86 ≈ 7.67 m/s. No need to know the ramp angle or length.