Kinetic energy
Kinetic energy is the energy an object carries simply because it is moving. Strike a cue ball in a game of billiards and you feel the idea in your hands. The cue stick gives the ball energy. The ball rolls, collides, and slows. The ball it hits leaps forward in its place. Nothing here was destroyed. Something invisible was handed from one body to another, and physics gives that something a name.
The word kinetic comes from the Greek kinesis, meaning motion. But naming the thing took centuries of argument. Why did it take until the mid-19th century for the terms kinetic energy and work to settle into their modern scientific meanings? Why does the same speeding bullet carry energy for one observer and none at all for another? And why does doubling a car's speed not double its energy, but quadruple it? The answers run from a block of clay in 1722 to the operators of quantum mechanics.
Gottfried Leibniz and Johann Bernoulli described the quantity that survives in a moving body as the living force, or vis viva. They developed the principle that E proportional to m times v squared is conserved in classical mechanics. The idea sat opposite a second one. The split between kinetic energy and potential energy traces back to Aristotle's concepts of actuality and potentiality.
Willem 's Gravesande of the Netherlands turned the argument into an experiment in 1722. He dropped weights from different heights into a block of clay. The depth each weight sank told him something exact. Penetration depth was proportional to the square of the impact speed, not to the speed itself. Émilie du Châtelet recognized what the clay was saying and published an explanation of the experiment's implications.
Thomas Young, in his 1802 lecture to the Royal Society, was the first to use the word energy to mean kinetic energy in its modern sense, in place of vis viva. Gaspard-Gustave Coriolis laid out the mathematics in an 1829 paper titled Du Calcul de l'Effet des Machines. William Thomson, later Lord Kelvin, is credited with coining the term kinetic energy around 1849 to 1851. William Rankine, who had introduced the term potential energy in 1853 along with the phrase actual energy, later noted that Thomson and Peter Tait swapped the word kinetic in for actual.
Energy occurs in many forms. Chemical energy, thermal energy, electromagnetic radiation, gravitational energy, electric energy, elastic energy, nuclear energy, and rest energy all sort into two main classes. Potential energy is one. Kinetic energy, the movement energy of an object, is the other. The first can be passed between objects and turned into other kinds of energy entirely.
A cyclist makes the trade visible. Chemical energy from food becomes the kinetic energy of the rider and the bicycle as they speed up. On a level surface that speed holds without further work, except to fight air resistance and friction. The conversion is never perfectly efficient. Some of the chemical energy turns into thermal energy inside the cyclist's own body.
The same cyclist can spend that motion in several different ways. Coast up a hill until the bicycle halts, and the kinetic energy has largely become gravitational potential energy, ready to be released by freewheeling down the far side. Connect a dynamo to a wheel on the descent, and some of the motion becomes electrical energy, so the bicycle reaches the bottom slower than it otherwise would. Apply the brakes instead, and friction scatters the kinetic energy as heat. The energy is never destroyed. It only changes form.
Flywheels were developed as a way to store energy, showing that kinetic energy also lives in rotational motion. A spacecraft pushes the point further. It burns chemical energy to launch and gain the kinetic energy needed to reach orbital velocity. In a perfectly circular orbit that energy stays constant, because there is almost no friction in near-earth space. The cost comes due at re-entry, when some of that kinetic energy turns into heat.
Half the product of the mass and the square of the speed gives the kinetic energy of a non-rotating body in classical mechanics. Mass is measured in kilograms, speed in metres per second, and the result comes out in joules. The joule is the SI unit of energy. The English unit is the foot-pound. Calculate the figure for an 80 kg mass, about 180 pounds, moving at 18 metres per second, about 40 miles per hour or 65 kilometres per hour, and you get a single number in joules.
The square in the formula has consequences that surprise people. An object that doubles its speed gains four times the kinetic energy, not twice. A car going twice as fast as another needs four times the distance to stop under the same braking force. Doubling the speed therefore takes four times the work.
Work is the bridge between motion and energy. The kinetic energy of a moving object equals the work needed to bring it from rest to that speed. It is also the work the object can do while being brought back to rest. Net force times displacement equals kinetic energy. Throw a ball and your hand does work to give it speed. The ball then strikes something and does work on what it hits.
A bullet passing an observer carries kinetic energy in that observer's frame of reference. The same bullet, viewed by someone moving alongside it at the same velocity, sits perfectly still and has zero kinetic energy. Because kinetic energy is a function of velocity, it depends on the relationship between the object and the observer's frame. It is not invariant.
A system of objects behaves differently. Its total kinetic energy cannot be reduced to zero by any choice of inertial frame unless every object shares the same velocity. In every other case there is a non-zero minimum. That minimum belongs to the center of momentum frame, the frame in which the total momentum of the system is zero. This lowest value contributes to the invariant mass of the system as a whole.
The choice of frame even decides how a rocket's energy is shared out. The chemical energy a rocket engine converts to kinetic energy is divided differently between the ship and its exhaust stream depending on the frame chosen. This is called the Oberth effect. Yet the total energy of the isolated system, counting kinetic energy, fuel, and heat together, does not change over time. Different observers in different frames will simply disagree on the value of that conserved total.
A rigid body rotating about a line through its center of mass holds rotational kinetic energy. That energy is just the sum of the kinetic energies of all its moving parts. It depends on the body's angular velocity and its moment of inertia, taken about an axis through the center of mass. A tennis ball in flight carries the kinetic energy of its rotation plus the kinetic energy of its translation through the air.
Groups of bodies hold internal kinetic energy from their relative motion. In the Solar System the planets and planetoids orbit the Sun. In a tank of gas the molecules move in every direction at once. The kinetic energy of any such system is the sum of the kinetic energies of the bodies inside it.
Even a stationary macroscopic body hides motion within. At the molecular and atomic level there is energy from molecular translation, rotation, and vibration, from electron translation and spin, and from nuclear spin. All of these internal energies contribute to the body's mass, inertia, and total energy, as the special theory of relativity provides. When people speak of a moving object's kinetic energy, they usually mean only the macroscopic movement.
Fluids carry the idea into flow. In fluid dynamics, the kinetic energy per unit volume at each point in an incompressible flow field is called the dynamic pressure at that point. It depends on the density of the fluid. Collisions show the other edge of the concept. In elastic collisions, like those in billiards, kinetic energy is preserved. In inelastic collisions it is dissipated as heat, sound, and binding energy when bound structures break apart.
The classical formula for kinetic energy is only a good approximation when an object's speed is much less than the speed of light. Push the speed to a significant fraction of light, and relativistic mechanics takes over. There, energy combines with momentum the way time and space combine into spacetime. A body's energy can be written in terms of its rest mass, its speed, and c, the speed of light in vacuum.
Kinetic energy in relativity is the total energy minus the rest energy. A by-product of this calculation is the mass-energy equivalence formula, the statement that mass and energy are essentially the same thing. At low speeds the relativistic expression collapses back into the Newtonian one. Expand the square root using the binomial approximation, and the rest energy drops out, leaving the familiar classical kinetic energy.
The corrections at human speeds are tiny but measurable. At 10 kilometres per second, the relativistic correction to the non-relativistic kinetic energy is 0.0417 joules per kilogram, on a non-relativistic value of 50 megajoules per kilogram. At 100 kilometres per second the correction grows to 417 joules per kilogram, on a value of 5 gigajoules per kilogram. The lesson sits inside the math. The formulae for energy and momentum are not special axioms. They emerge from the equivalence of mass and energy together with the principles of relativity.
In quantum mechanics, kinetic energy stops being a number and becomes an operator. For a single particle of mass m, the kinetic energy operator appears as a term in the Hamiltonian, built from the more fundamental momentum operator. It can be obtained by replacing momentum with its operator form in the classical expression for kinetic energy. In the Schrodinger picture the operator involves derivatives taken with respect to position coordinates.
For a system of N electrons described by a wavefunction, the expected electron kinetic energy is a sum of one-electron expectation values. Each term involves the mass of the electron and the Laplacian operator acting on the coordinates of one electron, summed over them all. The energy is, in this view, spread across every electron at once.
The density functional approach asks for less. It formally requires only the electron density, not the wavefunction itself. Given that density, the exact N-electron kinetic energy functional is unknown. There is one clean exception. For a single-electron system the kinetic energy can be written down exactly as the von Weizsacker kinetic energy functional, the rare case where the density alone is enough.
Common questions
What is kinetic energy in physics?
Kinetic energy is the form of energy an object possesses due to its motion. In classical mechanics, the kinetic energy of a non-rotating body equals half the product of its mass and the square of its speed. It equals the work needed to accelerate the object from rest to its speed.
Who coined the term kinetic energy?
William Thomson, later Lord Kelvin, is credited with coining the term kinetic energy around 1849 to 1851. William Rankine, who introduced the term potential energy in 1853, later noted that Thomson and Peter Tait substituted the word kinetic for the earlier phrase actual energy.
Where does the word kinetic come from?
The adjective kinetic has its roots in the Greek word kinesis, meaning motion. The dichotomy between kinetic energy and potential energy can be traced back to Aristotle's concepts of actuality and potentiality.
How did Willem 's Gravesande prove the kinetic energy relationship?
Willem 's Gravesande provided experimental evidence in 1722 by dropping weights from different heights into a block of clay. He determined that their penetration depth was proportional to the square of their impact speed. Emilie du Chatelet recognized the implications and published an explanation.
Why does doubling an object's speed quadruple its kinetic energy?
Kinetic energy increases with the square of the speed, so an object doubling its speed has four times as much kinetic energy. As a consequence, a car traveling twice as fast needs four times the distance to stop under a constant braking force, and it takes four times the work to double the speed.
What is the SI unit of kinetic energy?
The SI unit of energy, including kinetic energy, is the joule, where mass is measured in kilograms and speed in metres per second. The English unit of energy is the foot-pound.
Why is kinetic energy not invariant between observers?
Because kinetic energy is a function of velocity, it depends on the observer's frame of reference. A bullet passing an observer has kinetic energy in that observer's frame, but the same bullet has zero kinetic energy for an observer moving alongside it at the same velocity.
All sources
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