Work, Energy, and Power

What this unit covers

The topics below follow the published Physics 1 course framework for Unit 3. This unit is worth 18–23% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Lessons in this unit

• Work & the Work–Energy Theorem13 min · 3 objectivesCompute work as W = Fd cosθ and identify its sign · Recognize when a force does zero work · Apply the work–energy theorem, W_net = ΔKE
• Kinetic & Potential Energy13 min · 3 objectivesCompute kinetic energy with KE = ½mv² · Compute gravitational potential energy with PE = mgh · Compute elastic potential energy stored in a spring, ½kx²
• Conservation of Energy14 min · 3 objectivesState the conservation of mechanical energy for frictionless systems · Convert between potential and kinetic energy in falls, ramps, and swings · Account for energy lost to friction as thermal energy
• Power12 min · 3 objectivesDefine power as the rate of doing work, P = W/t · Relate power to force and velocity, P = Fv · Compare machines by how quickly they deliver energy
• Systems, Non-Conservative Forces & Energy Bookkeeping16 min · 3 objectivesDefine a system boundary and explain how it determines whether a force does work or stores potential energy · Apply the work–energy theorem when friction or an applied force is present · Track energy through a multi-stage problem where mechanical energy is not conserved
• Energy Graphs & Variable Forces15 min · 3 objectivesFind work as the area under a force–displacement graph, including for non-constant forces · Read a potential-energy curve to locate equilibrium points and classify their stability · Use the fact that force is the negative slope of the potential-energy curve

Formulas in Unit 3

Every term in Unit 3

All 51 terms we publish for Work, Energy, and Power, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Conservation of mechanical energy

KE + PE is constant when only conservative forces do work. The condition is what makes the statement true, and omitting it is what loses the point.

Power

P = W/Δt = Fv cos θ, measured in watts. At constant speed the drive force equals the resistive force, so P = Fv.

Work from a graph

The area under a force-position graph. This is how work is found when the force is not constant, where W = Fd fails.

Kinetic energy

KE = ½mv². Doubling speed quadruples kinetic energy, which is why stopping distance grows so fast with speed.

Hooke's law

F = −kx. The negative sign shows the restoring force opposes the displacement.

Conservative vs non-conservative forces

Conservative forces (gravity, spring) store energy recoverably and do path-independent work; friction and drag dissipate it.

Conservation of mechanical energy condition

KE + PE is constant ONLY when non-conservative forces do no work. With friction present, use energy accounting including thermal energy instead.

Energy bar charts

Track energy between states, including thermal energy generated by friction, so that total energy is conserved even when mechanical energy is not.

Choosing energy or kinematics

Energy methods are best when force varies or the path is complex; kinematics needs constant acceleration and gives timing information energy cannot.

Sign of work

Positive when force has a component along the displacement, negative when opposed. Friction on a sliding block does negative work.

Work done by a spring

The area under the F vs x line, which is ½kx² since the force grows linearly. Using F = kx times x would double-count.

Energy accounting with friction

Initial energy = final energy + friction force × path length. Note it is path LENGTH, not displacement — friction is not conservative.

Choosing the reference height

Any height may be defined as zero potential energy; only differences matter. Pick the lowest point of the motion to keep terms positive.

Energy in a pendulum

All potential at the extremes, all kinetic at the bottom. Maximum speed occurs where potential energy is minimum.

Power in terms of velocity

P = Fv, so a car at constant speed against constant drag has power proportional to speed and does no net work.

Why energy methods ignore the path

Conservative forces do path-independent work, so only endpoints matter — which is what makes a curved frictionless track solvable without calculus.

Escape and binding energy qualitatively

An object bound in a gravitational well has negative total energy; supplying enough energy to reach zero lets it escape.

Work

W = Fd cos θ, the energy transferred by a force acting through a displacement. Zero when the force is perpendicular to the motion, however large the force.

Why carrying a box horizontally does no work

The upward force is perpendicular to the horizontal displacement, so cos 90° = 0. It is tiring, and tiring is not the physics definition of work.

Work from a force-distance graph

The area under the curve, with area below the axis negative. The route to the work done by a spring, whose force is not constant.

Work-energy theorem

W_net = ΔKE. The total work by all forces equals the change in kinetic energy, which is often faster than kinematics plus Newton's second law.

Gravitational potential energy

ΔPE = mgΔh near Earth's surface. Only the CHANGE is physical; where you put h = 0 is a free choice and must be stated.

Elastic potential energy

PE = ½kx², the energy stored in a spring displaced x from equilibrium. Quadratic, so twice the compression stores four times the energy.

Conservative vs nonconservative force

A conservative force stores recoverable energy and its work is path-independent — gravity, springs. A nonconservative force dissipates it — friction, drag.

When mechanical energy is not conserved

Whenever friction, drag or a collision dissipates energy. Total energy is still conserved; some has become thermal.

Power at constant speed

A car at steady speed against drag delivers P = Fv where F is the drag force. Since drag rises with speed, power required rises faster than speed.

Energy in a spring launcher

Elastic PE converts to kinetic energy: ½kx² = ½mv². Solving for v gives v = x√(k/m).

Speed at the bottom of a frictionless ramp

mgh = ½mv², so v = √(2gh). Mass cancels and the ramp shape does not matter — only the height drop.

Why mass cancels in energy problems

Both gravitational potential energy and kinetic energy are proportional to mass, so it divides out whenever the only energies are those two.

Work done by friction

W = −f·d, using the PATH LENGTH rather than displacement, because friction acts along the whole path traveled.

Internal (thermal) energy

Energy dissipated by friction and drag, which raises temperature. It is where the "missing" mechanical energy went.

Efficiency

Useful energy output divided by total energy input, as a percentage. Never above 100%, and an answer that is says a step was double-counted.

Why the system boundary decides your equation

Put Earth inside the system and gravity is internal, so you write ΔU. Put Earth outside and gravity is external, so you write work. Doing both double-counts; doing neither loses the term.

Conservative force, defined by path independence

The work it does depends only on start and end points, never on the route. Gravity and ideal springs qualify — which is precisely why each has a potential-energy function and friction does not.

Why friction has no potential energy

Drag a block in a circle back to its start and friction has done negative work the whole way. The work depends on path length, so no function of position can describe it.

Thermal energy from friction uses path length

ΔE_thermal = f_k · d, where d is the distance traveled along the surface, not the displacement. Sliding 3 m out and 3 m back generates heating over 6 m.

Zero-work forces

Any force perpendicular to the motion does no work: the normal force, the tension on an orbiting satellite, and the centripetal force in uniform circular motion. None of them changes kinetic energy.

Work from a force–displacement graph

The area under the curve, which is the general definition. W = Fd cos θ is only the special case where the force is constant.

Where ½kx² comes from

It is the triangle under the line F = kx, base x and height kx. Not a separate rule to memorize — the same area-under-the-graph statement.

Force is the negative slope of U(x)

F = −dU/dx. The minus sign means force always points downhill on the potential-energy curve, toward lower potential energy.

Stable vs unstable equilibrium

Both are flat points on the U curve where force is zero. A minimum is stable — displaced, the force pushes back, so it oscillates. A maximum is unstable — displaced, the force drives it further away.

Reading a turning point off a U curve

Where the total-energy line meets the curve, K = 0 and the object reverses. Regions where U exceeds E are simply unreachable.

When to use energy instead of forces

When the problem gives speeds and positions but no time, or when the force is not constant so the kinematic equations fail. If it asks for a force at one instant, use Newton instead.

Work is a scalar

It has a sign but no direction. Adding works from several forces is ordinary addition with signs, never vector addition.

Negative work removes energy

A force with a component opposite the displacement does negative work. Friction and a braking force do negative work, which is why the object slows.

A joule is a newton-meter

Work is force times displacement, so the units are identical. Torque is also newton-meters but is never written in joules, because it is not energy.

Kinetic energy is never negative

K = 1/2 mv^2, and squaring the velocity removes the sign. A negative kinetic energy always means a sign was carried into the square by mistake.

Doubling the speed quadruples the kinetic energy

K depends on v squared, so a car at 60 carries four times the kinetic energy of the same car at 30, and needs four times the braking distance at constant force.

A watt is a joule per second

Power is the rate of energy transfer. A 60 W bulb converts 60 J every second, and a bigger engine does the same work in less time rather than more work.

Energy is conserved even when mechanical energy is not

Friction does not destroy energy, it moves it into thermal energy. Say mechanical energy is not conserved; saying energy is not conserved is wrong.

Power is the slope of an energy-time graph

Steeper means more power. A flat section means energy is not being transferred, whatever the object is doing.

What examiners penalize here

• Before computing work, check the angle between the force and the displacement. Perpendicular forces (normal force on a flat surface, gravity in horizontal motion, tension in circular motion) do exactly zero work — a favorite AP trap.
• You may place the h = 0 reference wherever it is most convenient — the floor, the tabletop, the ground below a cliff. Only the *change* in height between start and finish affects the physics, so choose the level that makes the arithmetic simplest.
• If a problem mentions friction or air resistance, mechanical energy is *not* conserved — the "missing" energy became heat. Write KE_i + PE_i = KE_f + PE_f + E_thermal and treat the thermal term as the energy removed.
• Remember that power and energy are different quantities: energy (joules) is the total transferred, power (watts) is the rate. A question asking "how quickly" or "per second" is about power; "how much total" is about energy or work.
• On a free-response, state your system and your zero of potential energy in words before writing any equation. Both are rubric points on energy questions, and choosing the ground as U = 0 makes almost every problem arithmetically cleaner.
• When a graph question gives a force in newtons and a position axis in centimeters, convert before computing the area. Unit slips on graph-area questions are more common than conceptual errors, and the rubric does not distinguish between the two.

Practice Physics 1

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

Questions about this unit

All 8 units of AP Physics 1: Algebra-Based

1. Unit 1 · Kinematics
2. Unit 2 · Force and Translational Dynamics
3. Unit 3 · Work, Energy, and Power
4. Unit 4 · Linear Momentum
5. Unit 5 · Torque and Rotational Dynamics
6. Unit 6 · Energy and Momentum of Rotating Systems
7. Unit 7 · Oscillations
8. Unit 8 · Fluids