Energy is the bookkeeping that makes physics feel complete. Momentum tells you what changes; energy tells you why the change had to happen, and what the alternatives were. Two quantities do most of the work in mechanics: kinetic energy for motion, potential energy for position and configuration.
The two formulas
Kinetic energy — the energy of motion, rising with mass and with the square of speed:
KE = ½mv²
Gravitational potential energy — the energy of height, relative to some chosen zero:
PE = mgh where g ≈ 9.81 m/s²
Their sum is the mechanical energy. The kinetic energy calculator returns all three, so you can see where an object stands between the two extremes.
Worked example: a 10 kg object at 5 m/s and 20 m up
KE = ½ × 10 × 25 = 125 J.
PE = 10 × 9.81 × 20 = 1,962 J.
Total = 2,087 J — of which only about 6% is kinetic. Released, the object converts its 1,962 J of potential energy into kinetic as it falls. It arrives at the reference level at v = √(2gh) = √392 ≈ 19.8 m/s, with the kinetic energy exactly matching the potential it lost.
That √(2gh) is a workhorse. Notice what it does not contain: mass. A feather and a brick fall at the same speed in a vacuum because both convert the same g·h into the same speed.
Why the square matters more than the mass
Doubling speed quadruples kinetic energy: ½m(2v)² = 4 × ½mv². Doubling mass only doubles it. This is the physics behind an uncomfortable number — aerodynamic drag rises roughly with the square of speed, so the power to hold a speed rises with the cube.
It explains why motorway fuel costs rise so steeply with speed, why a cyclist at 30 km/h feels roughly eight times the wind resistance of one at 15, and why terminal velocity exists at all: as speed grows, drag grows until it balances the driving force. The velocity calculator makes it easy to see what speed a fall produces from a given height.
Conservation, and its real meaning
The conservation law is not "nothing ever changes" — that is a misreading that makes people think conservation is a constraint rather than a permission. It says that in an isolated system with no friction and no drag, the total is fixed while the split between KE and PE can move freely. A pendulum swings up: KE falls, PE rises. It reaches the top with KE near zero, then reverses, and the same energy carries it back. The total never budges.
The useful consequence is height–speed exchange: if you know the total, you can predict either half from the other. A ball released at 20 m cannot be moving faster than 19.8 m/s at the bottom, no matter how it was dropped, because the energy simply is not there. That single fact explains a surprising amount of sports and safety physics.
What happens in the real world
Every actual system is lossy. Air resistance turns KE into heat and turbulence; friction at a pivot dissipates energy; a deforming ball and a soft surface convert some into sound and permanent deformation; internal friction warms the object. That is why a real pendulum's amplitude decays steadily — each swing is a little shorter, the energy going into the air and the pivot, until the swing stops.
The energy is not destroyed, only converted into forms the pendulum no longer stores. That distinction matters in engineering: if you want a ball to bounce back to its original height on a real surface, you must supply more than the energy that landed, because the loss is unavoidable.
Springs and a change of shape
Elastic potential energy stores work in a deformation. Hooke's law, F = kx, gives the force; integrating gives the stored energy:
E = ½kx²
The same square as kinetic energy, and the same lesson: double the compression and you quadruple the stored energy. Springs, rubber bands, compressed gas and a bent bow all store energy this way, and releasing it produces exactly the motion the conservation law predicts.
Power: energy per unit time
Power is the rate at which energy moves: P = E ÷ t, in watts. The kinetic-energy difference over time gives the power required to change speed. A 1,200 kg car from 0 to 25 m/s in 8 s needs KE = ½ × 1200 × 625 = 375,000 J, so P = 375,000 ÷ 8 ≈ 47 kW — before any losses. This is why a heavy car needs a large brake too: stopping returns that same energy, and the only question is where it goes.
Units: joules, calories and the food label
Energy in SI is measured in joules — one joule is the work done moving one kilogram one metre per second squared. Two other units dominate everyday life, and knowing the conversions prevents a common order-of-magnitude mistake.
A calorie (lowercase, the thermochemical one) is 4.184 J: the energy to raise 1 g of water by 1 °C. A kilocalorie, the unit on food labels and the "calories" in fitness talk, is 4,184 J. So a 500 kcal meal is about 2.09 million joules, and a 100 W light bulb running for an hour uses 360,000 J — roughly 86 kcal. That comparison is one of the more effective ways to see what an energy figure means physically.
One important caveat about food energy: the "calories" on a label are metabolically derived, not measured by combustion. The body digests protein and fibre with incomplete efficiency, so the metabolisable energy is smaller than the calorific value, which is why a food can advertise 100 kcal and deliver fewer usable calories. This is thermodynamics with a digestive system attached — energy is conserved, but the fraction you actually get depends on the chemistry of what you ate.
Efficiency, and why no machine is perfect
No real process converts energy perfectly. An electric motor typically reaches 85–95% efficiency, an internal combustion engine 20–40%, a light bulb a few percent. The remainder becomes heat, and that heat is not a failure — it is a consequence of the second law of thermodynamics: no process is completely free of friction, and energy always ends up as dispersed heat.
This is why "efficient" machines still get warm, and why the second law has practical teeth: you cannot have a process that converts 100% of input energy into useful work, and the maximum depends on the temperatures involved. The remaining energy has to go somewhere, and "somewhere" is always a lot of heat spread thinly into the surroundings.
Frequently asked questions
What is the formula for kinetic energy?
KE = ½mv², with m in kilograms and v in metres per second, giving joules. Because of the square, doubling the speed quadruples the kinetic energy — which is why small speed increases cost so much more fuel.
How is potential energy calculated?
Gravitational PE = mgh — mass in kg, g ≈ 9.81 m/s², and height in metres above the chosen reference level. The answer depends on where you place height zero, though the differences between heights are what matter.
Is energy conserved when something falls?
In an ideal system with no air resistance, yes: the gravitational potential energy lost becomes kinetic energy gained, so the total is constant. With air resistance some becomes heat and sound, so total mechanical energy decreases.
Why does energy conservation hold even when it appears to fail?
It does not fail. In a real system some mechanical energy becomes heat, sound and deformation — it is still energy, just not in a form the system stores. Choosing the system boundary is what decides whether the mechanical total looks constant.