Work, energy and power: release the cart and watch the energy move

How to use this simulation

1. Release the cart with friction at zero first: potential energy on the left turns into kinetic energy, climbs back to potential energy on the other side, and the total-energy bar (E) stays exactly as tall the whole time.
2. Turn friction up: a little heat (Q) appears on every pass, and the cart can no longer climb back to its starting height.
3. Grab the cart directly and drag it up or down the left slope: the release height changes, and the slider tracks the same value.
4. With friction on, let the cart run until it settles at the bottom, then read “Average friction force” and “Average power” in the panel — both are the heat (Q) divided by distance and by time.
5. Double the mass: both potential and kinetic energy double, but the speed readout stays the same — mass does not change speed here, only how much energy is involved.

Walking versus running up the same stairs

Two friends carry identical bags up the same five flights of stairs. One walks up slowly, the other runs. Both climb the same height carrying the same load, so both do exactly the same amount of work. The one who ran is more out of breath, because he did that same work in less time. That "doing it faster" is exactly what power measures.

Here is a stranger question: does holding a heavy bookshelf overhead while standing still count as work? By the physics definition, no. A force can act on an object all day, but if the object does not move, that force does zero work — the everyday sense of "I worked hard" and the physics sense of "work" are not the same thing.

And a familiar sight: drop a ball from a height and it speeds up all the way down. The moment it is released it has zero speed and so zero kinetic energy, yet being high up gives it a kind of stored energy called potential energy. As it falls, that stored energy turns into kinetic energy. This page’s simulation shows exactly that trade on a pair of live bars.

Starting from zero: what work really means

A force does work on an object when the object moves, and moves at least partly in the direction the force points. The amount of work equals the force multiplied by the part of the displacement that lies along the force.

When the force points exactly along the displacement, the formula is simple: work equals force times distance, W = Fs. F is measured in newtons (N) and s in metres (m), and the unit of work is the newton-metre, given its own name, the joule (J). One joule is the work done by a force of one newton moving an object one metre in its own direction.

When the force makes an angle θ with the displacement, only the component of the force along the motion does any work, so the general formula is W = Fs cosθ. At θ = 0°, cosθ = 1 and it reduces to the simple case. At θ = 90°, cosθ = 0 and the work is zero — which is exactly why carrying a bag at head height across a level floor does zero work by that lifting force: the force is vertical, the walk is horizontal.

Work can be positive, negative or zero. It is positive when the force and the displacement point the same way (pushing a trolley forward). It is negative when the force opposes the motion (friction acting against a sliding block, taking kinetic energy away). It is zero whenever the force is exactly perpendicular to the motion.

Key terms at a glance

Get the vocabulary straight before the formulas; definition questions in exams come straight from this table.

What energy is, and its two mechanical forms

Energy is the capacity a body has to do work. It comes in many forms — light, heat, chemical, electrical — but this page is mainly about two mechanical forms: kinetic energy and potential energy.

Kinetic energy is the energy a body has because it is moving. A body at rest has zero kinetic energy; as speed grows, kinetic energy grows too, but not in step with speed — it grows with the square of speed. The formula is Eₖ = ½mv².

Potential energy is energy stored in a body because of where it is. Gravitational potential energy depends on the body’s mass, the acceleration due to gravity, and its height above a chosen reference level. The formula is Eₚ = mgh.

The sum of the two is called mechanical energy, E = Eₖ + Eₚ. With no friction or air resistance, this sum never changes — only the split between Eₖ and Eₚ shifts back and forth. That is the law of conservation of mechanical energy, covered below.

Where the formulas come from: deriving Eₖ and Eₚ

Both formulas follow from the work–energy theorem: the total work done on a body equals the change in its kinetic energy, W = ΔEₖ.

Power: the rate of doing work

Two people can do exactly the same work, yet the one who finishes sooner is said to be more powerful. Power measures the rate of doing work — how much work happens per unit time.

The formula is P = W/t. With W in joules and t in seconds, P comes out in joules per second, given its own unit, the watt (W). One watt is one joule of work every second. Bigger machines are rated in kilowatts (1,000 W), and the older unit horsepower still shows up in car and motorbike advertising: 1 hp ≈ 746 W.

For a constant force F moving a body at constant speed v, in a short time Δt the displacement is Δs = vΔt, so power P = W/Δt = FΔs/Δt = Fv. This shortcut solves a lot of power problems without ever finding time or distance separately, especially for cars and motors moving at a steady speed.

Conservation of energy, and what friction does to it

The law of conservation of energy says energy is never created or destroyed, only changed from one form to another. With no friction or air resistance, mechanical energy E = Eₖ + Eₚ never changes. Set the friction slider to zero in the simulation and this is exactly what plays out: as the cart rolls down, Eₚ falls and Eₖ rises by exactly the same amount, and the total-energy bar (E) stays the same height throughout.

Friction changes the picture. A friction force acts against motion, so it steadily takes some mechanical energy away and turns it into heat — the same reason your hands warm up when you rub them together. That heat can be written Q = f × s, where f is the friction force and s the distance travelled. The accounting becomes: starting mechanical energy = ending mechanical energy + heat produced — nothing was lost, it only changed form.

The simulation’s "average friction force" and "average power" readings come from that same heat Q: F = Q/s (heat ÷ distance) and P = Q/t (heat ÷ time). That is how W = Fs and P = W/t come alive in one demonstration, because Q is itself a kind of work — the work done against the friction force.

Try these in the simulation

Guess each result before you run it, then check.

• Release the cart from 5 m with friction at zero, then read the speed at the bottom. It should match v = √(2gh) = 9.90 m/s.
• Set friction to 0.05 and let the cart swing back and forth several times. Each swing reaches a lower height than the last, while the heat bar (Q) never shrinks, only grows.
• Push friction to its maximum, 0.30, and wait for the cart to settle. Then read "average friction force" and "average power" — both computed from the heat by dividing by distance and by time.
• Double the mass. Both potential and kinetic energy double, but the speed reading is unchanged — mass does not change the speed here (same as free fall), only the amount of energy involved.
• Drag the cart to a lower start height and then a higher one. The peak of the potential-energy bar changes each time, because Eₚ is proportional to the starting height.

Solved problems

Write down what is given, then the formula, then substitute. That order earns full marks in an exam too.

At a glance: everyday power ratings

These rounded, typical figures give a sense of how much power ordinary appliances draw. The motorbike’s power is also shown in horsepower, since that unit is still common in vehicle advertising.

Common mistakes

Avoiding these keeps marks from slipping away on work, energy and power problems.

• Confusing the everyday sense of "working hard" with the physics sense of work. Holding something heavy and still is tiring but does zero work, because there is no displacement.
• Forgetting that a force perpendicular to the motion does zero work, as with carrying a bag at head height across level ground.
• Dropping the square in the kinetic-energy formula. Doubling speed quadruples kinetic energy; it does not just double it.
• Stating potential energy without naming the reference level. The same object’s potential energy differs depending on whether it is measured from the floor or from a tabletop.
• Treating power and energy as the same thing. Energy is the total capacity to do work; power is how fast that capacity gets spent.
• Assuming mechanical energy stays constant even with friction present. Friction converts mechanical energy into heat; the mechanical energy alone decreases, though the total (mechanical plus heat) is conserved.
• Mixing up the watt and the watt-hour. The watt is a unit of power (a rate); the watt-hour or kilowatt-hour is a unit of energy — the one that shows up on an electricity bill.

Work, energy and power in real life

These ideas are not confined to the exam hall — the same rules run quietly behind many everyday machines.

• Hydroelectric dams: water stored high behind a dam has potential energy that turns into kinetic energy as it falls, spinning turbines that convert it into electrical energy.
• Car brakes: kinetic energy turns into heat through friction at the brake pads, which is why repeated hard braking can make brakes noticeably hot.
• Electricity bills: one "unit" of electricity is the energy used by a one-kilowatt appliance running for one hour. A higher-power appliance uses more energy in the same time.
• Roller coasters: a motor lifts the cars up the first hill, storing potential energy; the rest of the ride simply trades that energy back and forth between kinetic and potential, losing a little to friction each pass.
• Wind-up clocks: winding the key stores potential energy in a spring, which slowly turns into the kinetic energy that moves the hands.
• Cranes: a crane that can lift the same load in less time is doing the same work faster — that is exactly what makes it more powerful.

Exam corner

Definition questions usually ask for work, kinetic energy, potential energy and power stated precisely, including their units. Numerical questions mix work, both forms of energy and power in one problem, and often expect the work–energy theorem rather than a direct formula. Always state the reference level when quoting a potential energy, and check units before submitting a final answer — a power in joules or an energy in watts is an instant flag to an examiner.

Revision: the last-minute summary

One pass over this list the night before the exam is usually enough.

• Work: W = Fs when the force is along the motion, generally W = Fs cosθ. Unit: joule (J).
• Kinetic energy: Eₖ = ½mv². Potential energy: Eₚ = mgh. Mechanical energy: E = Eₖ + Eₚ.
• Power: P = W/t = Fv at constant speed. Unit: watt (W); 1 hp ≈ 746 W.
• With no friction or air resistance, mechanical energy is conserved; with friction, some of it becomes heat (Q), but the total (mechanical + heat) is still conserved.
• In the simulation, average friction force F = Q/s and average power P = Q/t — both computed straight from the heat, exactly as the formulas say.
• Work is zero whenever the force is perpendicular to the motion, or whenever there is no motion at all — however large the force is.

Frequently asked questions