Conservation of Energy Study Pack

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Last updated May 27, 2026

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Conservation of Energy Study Guide

Master the law of conservation of energy by working through kinetic and potential energy (KE = ½mv², PE = mgh), the work-energy theorem, and the role of nonconservative forces like friction.

Key Takeaways

  • •The law of conservation of energy states that the total mechanical energy of an isolated system — the sum of kinetic energy and potential energy — remains constant as long as only conservative forces act on it.
  • •Kinetic energy (KE = ½mv²) depends on an object's mass and speed, while gravitational potential energy (PE = mgh) depends on mass, gravitational acceleration, and height above a reference point.
  • •When nonconservative forces such as friction or air resistance act on a system, mechanical energy is not conserved — some is converted to thermal energy or sound, but total energy (including all forms) is still conserved.
  • •The work-energy theorem connects net work done on an object to its change in kinetic energy: W_net = ΔKE, providing a bridge between force, displacement, and energy.
  • •Energy transformations — such as a roller coaster converting potential energy to kinetic energy and back — follow predictable patterns governed by conservation laws, allowing unknown speeds or heights to be calculated algebraically.
  • •Work done by a conservative force (like gravity or a spring) can be fully recovered as kinetic energy, while work done by a nonconservative force permanently removes mechanical energy from the system.
  • •Power measures the rate at which energy is transferred or transformed (P = W/t = Fv), distinguishing how quickly the same amount of work is accomplished.

Forms of Mechanical Energy

Mechanical energy encompasses the two forms of energy most directly linked to an object's motion and position within a gravitational or elastic field, and understanding each form is essential before examining how they interconvert.

Kinetic Energy (KE)

  • •Kinetic energy is the energy an object possesses because of its motion, defined as KE = ½mv², where m is mass in kilograms and v is speed in meters per second.
  • •Because speed is squared, doubling an object's speed quadruples its kinetic energy — a nonlinear relationship with important practical consequences in vehicle safety and projectile analysis.
  • •KE is always a non-negative scalar quantity; direction of motion does not affect its magnitude.

Gravitational Potential Energy (PE_grav)

  • •Gravitational potential energy is stored energy resulting from an object's position in a gravitational field, calculated as PE = mgh, where g ≈ 9.8 m/s² near Earth's surface and h is height above a chosen reference level.
  • •The reference level (h = 0) is arbitrary — only changes in height matter when calculating changes in gravitational PE, so the reference can be set wherever is most convenient for a given problem.
  • •Increasing an object's height increases its gravitational PE proportionally; this stored energy can later be released as kinetic energy.

Elastic Potential Energy (PE_elastic)

  • •Elastic potential energy is stored in a deformed elastic object — most commonly a spring — and is given by PE_elastic = ½kx², where k is the spring constant (N/m) and x is the displacement from the spring's natural length.
  • •Like gravitational PE, elastic PE is fully recoverable: a compressed or stretched spring releases all stored energy as kinetic energy when nonconservative forces are absent.

Work and the Work-Energy Theorem

Work is the mechanism by which energy is transferred to or from an object through the application of a force over a displacement, and the work-energy theorem formalizes the direct relationship between net work and changes in kinetic energy.

Definition of Work

  • •Work (W) is defined as W = Fd cosθ, where F is the magnitude of an applied force, d is the displacement of the object, and θ is the angle between the force vector and the displacement vector.
  • •Only the component of force parallel to the displacement does work; a force perpendicular to motion (like the normal force on a horizontally moving object) does zero work.
  • •Work is a scalar measured in joules (J); it can be positive (force aids motion), negative (force opposes motion), or zero.

The Work-Energy Theorem

  • •The work-energy theorem states that the net work done on an object equals its change in kinetic energy: W_net = KE_final − KE_initial = ΔKE.
  • •This theorem applies regardless of whether forces are conservative or nonconservative — it accounts for all forces collectively acting on the object.
  • •A net positive work increases an object's speed; net negative work (e.g., friction decelerating a sliding block) decreases it.

Conservative vs. Nonconservative Forces

  • •A conservative force does work that is path-independent — the work done moving an object between two points is the same regardless of the route taken. Gravity and spring forces are the primary examples.
  • •A nonconservative force, such as kinetic friction or air drag, does work that depends on the path length, and the energy it removes from mechanical motion cannot be recovered as mechanical energy — it disperses as heat or sound.

The Law of Conservation of Energy

The law of conservation of energy is one of the most fundamental principles in physics, establishing that energy cannot be created or destroyed — only converted from one form to another.

Conservation of Total Mechanical Energy

  • •When only conservative forces act on a system, total mechanical energy E = KE + PE remains constant throughout the motion: KE_i + PE_i = KE_f + PE_f.
  • •This equality allows unknown quantities (such as the speed at the bottom of a ramp or the maximum height of a projectile) to be solved algebraically without needing to track forces at every instant.
  • •A freely falling object perfectly illustrates this principle: as height decreases and PE falls, KE rises by an equal amount, keeping E constant.

Energy Transformation in Practice

  • •On a frictionless roller coaster, total mechanical energy stays fixed — the car trades height for speed continuously, with maximum speed at the lowest point and maximum height at momentary stops.
  • •A pendulum swinging in a vacuum converts PE entirely to KE at the bottom of its arc and entirely back to PE at each peak, with no net energy gain or loss over a complete cycle.

Role of Nonconservative Forces in Real Systems

  • •When friction or air resistance is present, mechanical energy decreases over time: KE_i + PE_i = KE_f + PE_f + W_nonconservative, where W_nonconservative represents energy converted to non-mechanical forms.
  • •Total energy — including thermal energy generated by friction — remains conserved across the entire universe; mechanical energy alone is not.
  • •Engineers use this principle to calculate energy losses in machines, allowing efficiency (useful energy output / total energy input) to be quantified and improved.

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