Conservation of energy states that energy is never created or destroyed, only converted from one form into another, so the total in a closed system stays constant. In mechanics that means kinetic energy plus potential energy holds steady, ½mv² + mgh, as long as nothing leaks away as heat, sound or deformation.
Let go of a ball at the top of a slide and it arrives at the bottom moving fast. Carry it back up and you pay for every metre yourself. Nothing in that exchange is free, and nothing in it vanishes.
That symmetry is the whole of this topic. Height turns into speed and speed turns back into height, and once you can count what went in you already know what comes out — no forces, no timings, no calculus.
What Is Conservation of Energy?
Conservation of energy is the rule that the total energy of a closed system never changes: energy moves between forms and between objects, but the books always balance.
The word “closed” is carrying most of the weight in that sentence. Draw a boundary around whatever you are studying, and the rule says the total inside it is fixed unless something crosses the line. Draw the boundary carelessly and the law will look broken when it is only being misread.
In mechanics the accounting usually runs to two entries, both of them ordinary forms of energy. Kinetic energy is what a body has because it is moving; potential energy is what it has because of where it sits. Slide down a hill and the second becomes the first.
Three things have to be true before you can use the rule to get a number.
- You have named the system. A ball alone, or a ball and the Earth, or a ball, the Earth and the track — the answer depends on which you mean, so say it before you start.
- You have a start state and an end state. Energy balances compare two snapshots. What happened between them is exactly what you are allowed not to know.
- You have accounted for anything that left. Friction, drag, a squashy impact — each of these moves energy out of the mechanical account, and pretending otherwise is much the commonest source of a wrong answer in this topic.
Why does this beat another formula? Because it lets you skip the middle. Work and forces want to know what is happening at every instant; an energy balance wants only the two ends of the journey, which is why an introductory course such as MIT’s OpenCourseWare classical mechanics course teaches force and the conservation laws side by side as two routes to the same answers.
One number recurs below. Take g = 9.81 m/s2 throughout — the conventional surface value rounded to three figures, and the figure the simulator on this page uses. Real local gravity runs from roughly 9.78 to 9.83 m/s2, so treat three significant figures as the ceiling on every answer here.
The Conservation of Energy Formula
For a body moving under gravity with nothing lost, the kinetic and potential energy at the start add up to the kinetic and potential energy at the end.
Real tracks are not frictionless. The honest version carries one more term on the right — everything that left the mechanical account along the way.
| Symbol | What it is | SI unit | Worth knowing |
|---|---|---|---|
| m | mass of the body | kg | divides out of every term when Eloss is zero, and only then |
| v1, v2 | speed at the start and at the end | m/s | speeds, not velocities; the equation is blind to direction |
| h1, h2 | height at the start and at the end | m | only the difference h1 – h2 ever enters, so the zero is yours to pick |
| g | gravitational field strength | m/s2 | 9.81 m/s2 throughout this page; the nominal lunar surface value is about 1.62 m/s2, so g is an input rather than a constant of nature |
| Eloss | mechanical energy that left the account | J | heat, sound, deformation; it can be zero but never negative |
Working that balance by hand is four lines of algebra every single time. The conservation of energy calculator takes the seven quantities in the table above — the mass, both speeds, both heights, the energy lost and g — and solves the same equation for whichever one you nominate: either speed, either height, the energy lost to friction, or the mass. That loss term is the piece the kinematics tools do not carry.
One habit saves more marks here than any other. Fix the zero of height before you write anything down, then measure both heights from it and never move it again. Because only h1 – h2 appears, the floor, the bench top and sea level are all equally valid choices — but only if you stick to one.
The simulator below is that balance with the algebra taken out. One slider sets the release height of a 1 kg ball in a frictionless valley, and four readouts track its kinetic energy, its potential energy, the total and its speed. The same sim runs on its own page as the conservation of energy simulator.
How the Energy Swap Actually Works
The swap works because gravity does the same amount of work on a falling body whichever way the trade is running. Fall through a height h and gravity does mgh of work on you; that work has to show up somewhere, and with nothing to rub against it shows up as speed.
Follow a 1 kg ball down a 3.00 m drop and the accounting is easy to watch. It begins with 29.43 J of potential energy and no motion at all. A third of the way down it has converted 9.81 J, so it holds 19.62 J of potential and 9.81 J of kinetic energy — and at the bottom the whole 29.43 J is kinetic.
Run the simulator above and the same trade plays out on screen. Two habits of the tool are worth knowing before they confuse you: its ball is a fixed 1 kg model sliding on a frictionless valley with g fixed at 9.81 m/s2, and every readout is rounded to one decimal place.
| Drop height | Kinetic energy, at rest | Potential energy, at rest | Total energy | Speed at the valley floor |
|---|---|---|---|---|
| 0.5 m | 0.0 J | 4.9 J | 4.9 J | 3.1 m/s |
| 1.2 m | 0.0 J | 11.7 J | 11.8 J | 4.9 m/s |
| 2.0 m | 0.0 J | 19.6 J | 19.6 J | 6.3 m/s |
| 3.0 m | 0.1 J | 29.4 J | 29.4 J | 7.7 m/s |
| 5.0 m | 0.1 J | 49.0 J | 49.1 J | 9.9 m/s |
Three details in that table repay a sentence each. The 1.2 m row prints 0.0 J and 11.7 J beside a total of 11.8 J, because each of the three numbers is rounded on its own — the quantities add up perfectly, the printed strings need not.
The 3.0 m and 5.0 m rows read 0.1 J before anything has been released, because the ball is parked a hair off the release point so that it has somewhere to start sliding from. A clean zero on the first frame is the wrong thing to expect here.
And whether a printed ratio lands cleanly depends on which readout you take it from. Quadrupling the height quadruples the energy and doubles the speed as quantities, but one decimal place does not always carry that across. Both energy columns run 4.9 J to 19.6 J between the 0.5 m and 2.0 m rows; at 0.9 m and 3.6 m the potential-energy column reads 8.8 J then 35.2 J, while the total at 3.6 m reads 35.3 J. Do the scaling from the formula rather than from the screen.
Why the Mass Cancels, and When It Stops Cancelling
The mass cancels because every term in the lossless balance carries exactly one factor of m — and it stops cancelling the moment a fixed amount of energy is taken out.
Write the balance out and the point is hard to miss. In ½mv12 + mgh1 = ½mv22 + mgh2, dividing every term by m simply deletes it. A 50 g ball and a 50 kg boulder released from the same shelf, with nothing lost on the way down, arrive at the floor at the same speed, and problem 1 below works that case through without the mass ever appearing.
Now put a fixed loss into the same sum. Drop a ball 1.80 m and let 3.00 J go to drag: a 2.00 kg ball lands at 5.68 m/s while a 4.00 kg ball lands at 5.82 m/s, against 5.94 m/s for the ideal case. The heavier body arrives faster, because the same 3.00 J is a smaller slice of its larger store.
So “mass never matters” is half a rule. It is exact when nothing is lost and wrong whenever something is — which, outside a physics question, is always.
Why the Path Does Not Change the Answer
The route makes no difference to the arrival speed, because gravity’s work depends only on the change in height and not at all on how the body got there.
Drop a ball 1.80 m, roll it down a straight ramp through the same 1.80 m, or send it down a curved slide: all three reach the floor at 5.94 m/s. That is the most useful consequence in the whole topic, because it lets you answer questions about shapes you could never write a force equation for.
Equal speeds are not equal times, and exam questions love that gap. The vertical drop gets there first by a wide margin; the shallow slide dawdles. Energy tells you how fast a body arrives and never when.
Forces that behave this way have a name: conservative. Gravity is one, an ideal spring is another, and for both of them the work done depends on the endpoints alone. Friction is not conservative — which is precisely why it needs a term of its own on the right-hand side.
Where the Energy Goes When It Is Lost
Energy that leaves the mechanical account has not disappeared; it has become heat, sound and permanent deformation, three forms that height and speed cannot see.
A bouncing ball measures the loss for you, with no instruments at all. Drop one from 2.00 m, watch it rebound to 1.40 m, and 30.0 % of its mechanical energy has gone in a single impact lasting a tiny fraction of a second.
It is tempting to read the rebound ratio as the energy ratio. It is not. The coefficient of restitution for that bounce is a ratio of speeds, sqrt(1.40/2.00) = 0.837, while the energy ratio is 0.700 — energy goes as the square of speed, so the two figures can never agree.
Here is where the bigger statement of the law lives. Mechanical energy is conserved only in the frictionless idealisation; total energy, heat included, is conserved always. Write “energy is conserved” beside a frictionless number without that qualifier and the next real experiment will embarrass you.
Where You Meet Conservation of Energy
The pattern to watch for is a store being emptied into motion, or motion being banked back into a store.
- A roller coaster. The chain lift does the work once, at the start. Every hill after it is paid for out of that one deposit, which is why the hills get lower as the ride goes on.
- A pendulum or a swing. The string does no work at all, because its pull is always sideways to the motion, so the bob trades height for speed and back again on a fixed budget.
- A pumped-storage power station. Cheap overnight electricity pumps water uphill; at peak demand the same water runs back down through the turbines. The US Department of Energy’s pumped storage hydropower page describes the two-reservoir arrangement that makes it work.
- A drawn bow or a compressed spring. The store here is elastic potential energy, ½kx2 rather than mgh, but the balance is written exactly the same way.
- Regenerative braking. An electric car slowing down pushes much of its kinetic energy back into the battery instead of turning all of it into hot brake discs. Same law, better bookkeeping.
Notice what none of those five does. Not one of them creates energy; each only moves it between stores that were already stocked.
Common Misconceptions About Conservation of Energy
Four beliefs cause most of the trouble here, and each is worth naming out loud.
“We are running out of energy”
Not in the physicist’s sense. The energy in a tank of petrol is all still there after you burn it, spread thinly through warm exhaust, warm engine and warm road. What runs out is energy in a concentrated, useful form — the quantity is conserved, the usefulness is not.
“Mechanical energy is always conserved”
Only when nothing removes it. Air resistance, a rough track, a warm bearing and a squashy impact all move energy out of the mechanical account, so a real experiment always finds less kinetic plus potential energy at the end than at the start. The total is still conserved; it is the mechanical sub-total that leaks away, which is why a real pendulum eventually stops.
“Conservation of energy is the same rule as conservation of momentum”
They are two different conserved quantities with two different conditions: energy is a scalar in joules that holds steady when nothing crosses the system boundary, while conservation of momentum is a vector rule in kg·m/s that holds when no net external force acts. A collision can conserve momentum perfectly and still lose most of its kinetic energy to heat and sound.
“A heavier ball hits the ground harder, so it must be faster”
Harder, yes. Faster, only once you let the air in — with nothing lost they arrive together, because the mass divides out. Put a fixed loss in the sum and the heavier ball does arrive faster, as the drag figures further up show, and ½mv2 still contains an m, so it carries more kinetic energy either way. Speed and energy are separate questions, and this equation answers them separately.
