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Encyclopedia > Kinetic energy
The cars of a roller coaster reach their maximum kinetic energy when at the bottom of their path. When they start rising, the kinetic energy begins to be converted to gravitational potential energy, but the total amount of energy in the system remains constant; assuming negligible friction and other energy conversion factors.

The kinetic energy of an object is the extra energy which it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its current velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes. Negative work of the same magnitude would be required to return the body to a state of rest from that velocity. Image File history File links Size of this preview: 486 Ã— 600 pixelsFull resolutionâ€Ž (800 Ã— 987 pixels, file size: 208 KB, MIME type: image/jpeg) drue is a hoe The Texas Giant wooden roller coaster located at Six Flags over Texas. ... Image File history File links Size of this preview: 486 Ã— 600 pixelsFull resolutionâ€Ž (800 Ã— 987 pixels, file size: 208 KB, MIME type: image/jpeg) drue is a hoe The Texas Giant wooden roller coaster located at Six Flags over Texas. ... A typical roller coaster The roller coaster is a popular amusement ride developed for amusement parks and modern theme parks. ... For other uses, see Friction (disambiguation). ... In physics and engineering, energy conversion is any process of converting energy from one form to another. ... In physics, mechanical work is the amount of energy transferred by a force. ... Acceleration is the time rate of change of velocity and/or direction, and at any point on a velocity-time graph, it is given by the slope of the tangent to the curve at that point. ... Negative has meaning in several contexts: Look up negative in Wiktionary, the free dictionary. ...

The adjective "kinetic" to the noun energy has its roots in the Greek word for "motion" (kinesis). The terms kinetic energy and work and their present scientific meanings date back to the mid 19th century. Early understandings of these ideas can be attributed to Gaspard-Gustave Coriolis who in 1829 published the paper titled Du Calcul de l'Effet des Machines outlining the mathematics of kinetic energy. // Biology Kinesis is a movement or activity of a cell or an organism in response to a stimulus such as light. ... Gaspard-Gustave de Coriolis or Gustave Coriolis (May 21, 1792â€“September 19, 1843), mathematician, mechanical engineer and scientist born in Paris, France. ...

William Thomson, later Lord Kelvin, is given the credit for coining the term kinetic energy c. 1849.[citation needed] For other persons named William Thomson, see William Thomson (disambiguation). ...

## Introduction

Main article: Energy

There are various forms of energy : chemical energy, heat, electromagnetic radiation, potential energy (gravitational, electric, elastic, etc.), nuclear energy, rest energy. These can be categorized in two main classes: potential energy and kinetic energy. In chemistry, a chemical bond is the force which holds together atoms in molecules or crystals. ... For other uses, see Heat (disambiguation) In physics, heat, symbolized by Q, is energy transferred from one body or system to another due to a difference in temperature. ... This box:      Electromagnetic (EM) radiation is a self-propagating wave in space with electric and magnetic components. ... Potential energy can be thought of as energy stored within a physical system. ... This article concerns the energy stored in the nuclei of atoms; for the use of nuclear fission as a power source, see Nuclear power. ... The rest energy of a particle is its energy when it is not moving relative to a given inertial reference frame. ... Potential energy can be thought of as energy stored within a physical system. ...

Kinetic energy can be best understood by examples that demonstrate how it is transformed from other forms of energy and to the other forms. For example, a cyclist will use chemical energy that was provided by food to accelerate a bicycle to a chosen speed. This speed can be maintained without further work, except to overcome air-resistance and friction. The energy has been converted into the energy of motion, known as kinetic energy but the process is not completely efficient and heat is also produced within the cyclist. Potential energy can be thought of as energy stored within a physical system. ...

The kinetic energy in the moving bicycle and the cyclist can be converted to other forms. For example, the cyclist could encounter a hill just high enough to coast up, so that the bicycle comes to a complete halt at the top. The kinetic energy has now largely been converted to gravitational potential energy that can be released by freewheeling down the other side of the hill. (Since the bicycle lost some of its energy to friction, it will never regain all of its speed without further pedaling. Note that the energy is not destroyed; it has only been converted to another form by friction.) Alternatively the cyclist could connect a dynamo to one of the wheels and also generate some electrical energy on the descent. The bicycle would be traveling more slowly at the bottom of the hill because some of the energy has been diverted into making electrical power. Another possibility would be for the cyclist to apply the brakes, in which case the kinetic energy would be dissipated through friction as heat energy. For other uses, see Bicycle (disambiguation). ... A cyclist is a person who engages in cycling whether as a sport or rides a bicycle for recreation or transportation. ... This article is about machines that produce electricity. ...

Like any physical quantity which is a function of velocity, the kinetic energy of an object does not depend only on the inner nature of that object. It also depends on the relationship between that object and the observer (in physics an observer is formally defined by a particular class of coordinate system called an inertial reference frame). Physical quantities like this are said to be not invariant. The kinetic energy is co-located with the object and contributes to its gravitational field. In physics, an inertial frame of reference, or inertial frame for short (also descibed as absolute frame of reference), is a frame of reference in which the observers move without the influence of any accelerating or decelerating force. ...

### Calculations

There are several different equations that may be used to calculate the kinetic energy of an object. In many cases they give almost the same answer to well within measurable accuracy. Where they differ, the choice of which to use is determined by the velocity of the body or its size. Thus, if the object is moving at a velocity much smaller than the speed of light, the Newtonian (classical) mechanics will be sufficiently accurate; but if the speed is comparable to the speed of light, relativity starts to make significant differences to the result and should be used. If the size of the object is sub-atomic, the quantum mechanical equation is most appropriate. It has been suggested that this article or section be merged with Classical mechanics. ... For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ... Fig. ...

## Newtonian kinetic energy

### Kinetic energy of rigid bodies

In classical mechanics, the kinetic energy of a "point object" (a body so small that its size can be ignored), or a non rotating rigid body, is given by the equation $E_k = begin{matrix} frac{1}{2} end{matrix} mv^2$ where m is the mass and v is the velocity of the body. Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. ...

For example - one would calculate the kinetic energy of an 80 kg mass traveling at 18 meters per second (40 mph) as $begin{matrix} frac{1}{2} end{matrix} cdot 80 cdot 18^2 = 12,960 mathrm{joules}$.

Note that the kinetic energy increases with the square of the speed. This means, for example, that if you are traveling twice as fast, you will have four times as much kinetic energy. As a result of this, a car traveling twice as fast requires four times as much distance to stop (assuming a constant braking force. See mechanical work). In physics, mechanical work is the amount of energy transferred by a force. ...

Thus, the kinetic energy can be calculated using the formula:

$E_k = frac{1}{2}mv^2$

where:

Ek is the kinetic energy in joules
m is the mass in kilograms, and
v is the velocity in meters per second.

For the translational kinetic energy of a body with constant mass m, whose center of mass is moving in a straight line with speed v, as seen above is equal to The joule (IPA: or ) (symbol: J) is the SI unit of energy. ... For other uses, see Mass (disambiguation). ... In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the systems mass behaves as if it were concentrated. ...

$E_t = begin{matrix} frac{1}{2} end{matrix} mv^2$

where:

m is mass of the body
v is speed of the center of mass of the body.

Thus kinetic energy is a relative measure and no object can be said to have a unique kinetic energy. A rocket engine could be seen to transfer its energy to the rocket ship or to the exhaust stream depending upon the chosen frame of reference. But the total energy of the system, i.e. kinetic energy, fuel chemical energy, heat energy etc, will be conserved regardless of the choice of measurement frame. In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the systems mass behaves as if it were concentrated. ...

$E_k = frac{p^2}{2m}$

### Derivation and definition

The work done accelerating a particle during the infinitesimal time interval dt is given by the dot product of force and displacement:

$mathbf{F} cdot d mathbf{x} = mathbf{F} cdot mathbf{v} d t = frac{d mathbf{p}}{d t} cdot mathbf{v} d t = mathbf{v} cdot d mathbf{p} = mathbf{v} cdot d (m mathbf{v})$

Applying the product rule we see that:

$d(mathbf{v} cdot mathbf{v}) = (d mathbf{v}) cdot mathbf{v} + mathbf{v} cdot (d mathbf{v}) = 2(mathbf{v} cdot dmathbf{v})$

Therefore (assuming constant mass), the following can be seen:

$mathbf{v} cdot d (m mathbf{v}) = frac{m}{2} d (mathbf{v} cdot mathbf{v}) = frac{m}{2} d v^2 = d left(frac{m v^2}{2}right)$

Since this is a total differential (that is, it only depends on the final state, not how the particle got there), we can integrate it and call the result kinetic energy:

$E_k = int mathbf{F} cdot d mathbf{x} = int mathbf{v} cdot d mathbf{p}= frac{m v^2}{2}$

### Kinetic energy of systems

For a single point, or a rigid body that is not rotating, the kinetic energy goes to zero when the body stops.

However, for systems containing multiple independently moving bodies, which may exert forces between themselves, and may (or may not) be rotating; this is no longer true.

This energy is called 'internal energy'.

The kinetic energy of the system at any instant in time is simply the sum of the kinetic energies of the masses- including the kinetic energy due to the rotations.

An example would be the solar system. In the center of mass frame of the solar system, the Sun is (almost) stationary, but the planets and planetoids are in motion about it. Thus even in a stationary center of mass frame, there is still kinetic energy present. The center of mass frame (also called the center of momentum frame, CM frame, or COM frame) is defined as being the particular inertial frame in which the center of mass of a system of interest, is at rest (has zero velocity). ...

However, recalculating the energy from different frames would be tedious, but there is a trick. The kinetic energy of the system from a different inertial frame can be calculated simply from the sum of the kinetic energy in the center of mass frame and adding on the energy that the total mass of bodies in the center of mass frame would have if it were moving at the relative speed between the two frames.

This may be simply shown: let V be the relative speed of the frame k from the center of mass frame i :

$E_k = int frac{v_k^2 dm}{2} = int frac{(v_i + V)^2 dm}{2} = int frac{(v_i^2 + 2 v_i V + V^2) dm}{2} = int frac{v_i^2 dm}{2} + V int v_i dm + frac{V^2}{2} int dm$

However, let $int frac{v_i^2 dm}{2} = E_i$ the kinetic energy in the center of mass frame, $int v_i dm$ would be simply the total momentum which is by definition zero in the center of mass frame, and let the total mass: $int dm = M$. Substituting, we get:[1]

$E_k = E_i + frac{M V^2}{2}$

The kinetic energy of a system thus depends on the inertial frame of reference and it is lowest with respect to the center of mass reference frame, i.e., in a frame of reference in which the center of mass is stationary. In any other frame of reference there is an additional kinetic energy corresponding to the total mass moving at the speed of the center of mass. An inertial frame of reference, or inertial reference frame, is one in which Newtons first and second laws of motion are valid. ... In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the systems mass behaves as if it were concentrated. ...

### Rotating bodies

If a rigid body is rotating about any line through the center of mass then it has rotational kinetic energy (Er) which is simply the sum of the kinetic energies of its moving parts, and thus it is equal to: The rotational energy or angular kinetic energy is the kinetic energy due to the rotation of an object and is part of its total kinetic energy. ...

$E_r = int frac{v^2 dm}{2} = int frac{(r omega)^2 dm}{2} = frac{omega^2}{2} int{r^2}dm = frac{omega^2}{2} I = begin{matrix} frac{1}{2} end{matrix} I omega^2$

where:

ω is the body's angular velocity.
r is the distance of any mass dm from that line
I is the body's moment of inertia$= int{r^2}dm$

(In this equation the moment of inertia must be taken about an axis through the center of mass and the rotation measured by ω must be around that axis; more general equations exist for systems where the object is subject to wobble due to its eccentric shape). Angular velocity describes the speed of rotation and the orientation of the instantaneous axis about which the rotation occurs. ... Moment of inertia, also called mass moment of inertia or the angular mass, (SI units kg m2, Former British units slug ft2), is the rotational analog of mass. ...

### Rotation in systems

It sometimes is convenient to split the total kinetic energy of a body into the sum of the body's center-of-mass translational kinetic energy and the energy of rotation around the center of mass rotational energy: The rotational energy or angular kinetic energy is the kinetic energy due to the rotation of an object and is part of its total kinetic energy. ...

$E_k = E_t + E_r ,$

where:

Ek is the total kinetic energy
Et is the translational kinetic energy
Er is the rotational energy or angular kinetic energy in the rest frame

Thus the kinetic energy of a tennis ball in flight is the kinetic energy due to its rotation, plus the kinetic energy due to its translation.

## Relativistic kinetic energy of rigid bodies

In special relativity, we must change the expression for linear momentum. Integrating by parts, we get: For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ...

$E_k = int mathbf{v} cdot d mathbf{p}= int mathbf{v} cdot d (m gamma mathbf{v}) = m gamma mathbf{v} cdot mathbf{v} - int m gamma mathbf{v} cdot d mathbf{v} = m gamma v^2 - frac{m}{2} int gamma d (v^2)$

Remembering that $gamma = (1 - v^2/c^2)^{-1/2}!$, we get:

$E_k = m gamma v^2 - frac{- m c^2}{2} int gamma d (1 - v^2/c^2) = m gamma v^2 + m c^2 (1 - v^2/c^2)^{1/2} + C$

And thus:

$E_k = m gamma (v^2 + c^2 (1 - v^2/c^2)) + C = m gamma (v^2 + c^2 - v^2) + C = m gamma c^2 + C!$

The constant of integration is found by observing that $gamma = 1!$ when $mathbf{v }= 0$, so we get the usual formula:

$E_k = m gamma c^2 - m c^2 = frac{m c^2}{sqrt{1 - v^2/c^2}} - m c^2$

If a body's speed is a significant fraction of the speed of light, it is necessary to use relativistic mechanics (the theory of relativity as expounded by Albert Einstein) to calculate its kinetic energy. A line showing the speed of light on a scale model of Earth and the Moon, taking about 1â…“ seconds to traverse that distance. ... Albert Einsteins theory of relativity is a set of two theories in physics: special relativity and general relativity. ... â€œEinsteinâ€ redirects here. ...

For a relativistic object the momentum p is equal to:

$p = frac{m v}{sqrt{1 - (v/c)^2}}$,

where m is the rest mass, v is the object's speed, and c is the speed of light in vacuum. The term mass in special relativity is used in a couple of different ways, occasionally leading to a great deal of confusion. ...

Thus the work expended accelerating an object from rest to a relativistic speed is:

$E_k = frac{m c^2}{sqrt{1 - (v/c)^2}} - m c^2$.

The equation shows that the energy of an object approaches infinity as the velocity v approaches the speed of light c, thus it is impossible to accelerate an object across this boundary.

The mathematical by-product of this calculation is the mass-energy equivalence formula—the body at rest must have energy content equal to: 15ft sculpture of Einsteins 1905 E = mcÂ² formula at the 2006 Walk of Ideas, Germany In physics, mass-energy equivalence is the concept that all mass has an energy equivalence, and all energy has a mass equivalence. ...

$E_mbox{rest} = m c^2 !$

At a low speed (v<<c), the relativistic kinetic energy may be approximated well by the classical kinetic energy. This is done by binomial approximation. Indeed, taking Taylor expansion for square root and keeping first two terms we get: The binomial approximation is useful for approximately calculating powers of numbers close to 1. ... As the degree of the taylor series rises, it approaches the correct function. ...

$E_k approx m c^2 left(1 + frac{1}{2} v^2/c^2right) - m c^2 = frac{1}{2} m v^2$,

So, the total energy E can be partitioned into the energy of the rest mass plus the traditional Newtonian kinetic energy at low speeds.

When objects move at a speed much slower than light (e.g. in everyday phenomena on Earth), the first two terms of the series predominate. The next term in the approximation is small for low speeds, and can be found by extending the expansion into a Taylor series by one more term:

$E approx m c^2 left(1 + frac{1}{2} v^2/c^2 + frac{3}{8} v^4/c^4right) = m c^2 + frac{1}{2} m v^2 + frac{3}{8} m v^4/c^2$.

For example, for a speed of 10 km/s the correction to the Newtonian kinetic energy is 0.07 J/kg (on a Newtonian kinetic energy of 50 MJ/kg) and for a speed of 100 km/s it is 710 J/kg (on a Newtonian kinetic energy of 5 GJ/kg), etc.

For higher speeds, the formula for the relativistic kinetic energy [2] is derived by simply subtracting the rest mass energy from the total energy:

$E_k = m gamma c^2 - m c^2 = m c^2left(frac{1}{sqrt{1 - (v/c)^2}} - 1right)$.

$E_k = sqrt{p^2 c^2 + m^2 c^4} - m c^2$.

This can also be expanded as a Taylor series, the first term of which is the simple expression from Newtonian mechanics. Series expansion redirects here. ...

What this suggests is that the formulas for energy and momentum are not special and axiomatic, but rather concepts which emerge from the equation of mass with energy and the principles of relativity.

## Quantum mechanical kinetic energy of rigid bodies

In the realm of quantum mechanics, the expectation value of the electron kinetic energy, $langlehat{T}rangle$, for a system of electrons described by the wavefunction $vertpsirangle$ is a sum of 1-electron operator expectation values: The wave equation is an important partial differential equation which generally describes all kinds of waves, such as sound waves, light waves and water waves. ... A wave function is a mathematical tool that quantum mechanics uses to describe any physical system. ...

$langlehat{T}rangle = -frac{hbar^2}{2 m_e}bigglanglepsi biggvert sum_{i=1}^N nabla^2_i biggvert psi biggrangle$

where me is the mass of the electron and $nabla^2_i$ is the Laplacian operator acting upon the coordinates of the ith electron and the summation runs over all electrons. Notice that this is the quantized version of the non-relativistic expression for kinetic energy in terms of momentum: In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. ...

$E_k = frac{p^2}{2m}$

The density functional formalism of quantum mechanics requires knowledge of the electron density only, i.e., it formally does not require knowledge of the wavefunction. Given an electron density $rho(mathbf{r})$, the exact N-electron kinetic energy functional is unknown; however, for the specific case of a 1-electron system, the kinetic energy can be written as Density functional theory (DFT) is a quantum mechanical method used in physics and chemistry to investigate the electronic structure of many-body systems, in particular molecules and the condensed phases. ...

$T[rho] = frac{1}{8} int frac{ nabla rho(mathbf{r}) cdot nabla rho(mathbf{r}) }{ rho(mathbf{r}) } d^3r$

where T[ρ] is known as the Von Weizsacker kinetic energy functional.

## Some examples

Spacecraft use chemical energy to take off and gain considerable kinetic energy to reach orbital velocity. This kinetic energy gained during launch will remain constant while in orbit because there is almost no friction. However it becomes apparent at re-entry when the kinetic energy is converted to heat. The Space Shuttle Discovery as seen from the International Space Station. ... The orbital speed of a body, generally a planet, a natural satellite, an artificial satellite, or a multiple star, is the speed at which it orbits around the barycenter of a system, usually around a more massive body. ...

Kinetic energy can be passed from one object to another. In the game of billiards, the player gives kinetic energy to the cue ball by striking it with the cue stick. If the cue ball collides with another ball, it will slow down dramatically and the ball it collided with will accelerate to a speed as the kinetic energy is passed on to it. Collisions in billiards are effectively elastic collisions, where kinetic energy is preserved. This article is about the various cue sports. ... Physical collision Dynamics Deflection happens when an object hits a plane surface In physics, collision means the action of bodies striking or coming together (touching). ...

Flywheels are being developed as a method of energy storage (see article flywheel energy storage). This illustrates that kinetic energy can also be rotational. Note the formula in the articles on flywheels for calculating rotational kinetic energy is different, though analogous. Spoked flywheel Flywheel from stationary engine. ... Energy storage is the storing of some form of energy that can be drawn upon at a later time to perform some useful operation. ... NASA G2 flywheel Flywheel Energy Storage (FES) works by accelerating a rotor (flywheel) to a very high speed and maintaining the energy in the system as rotational energy. ...

 Energy Portal

Image File history File links Portal. ... An early naval cannon design, allowing the gun to roll backwards a small distance when firing The recoil when firing a gun is the backward momentum of a gun, which is equal to the forward momentum of the bullet or shell, due to conservation of momentum. ... The joule (IPA: or ) (symbol: J) is the SI unit of energy. ... The parallel axes rule can be used to determine the moment of inertia of a rigid object about any axis, given the moment of inertia of the object about the parallel axis through the objects center of mass and the perpendicular distance between the axes. ... Space Shuttle Atlantis launches on mission STS-71. ... A projectile is any object sent through space by the application of a force. ... A projectile is any object sent through space by the application of a force. ...

## Notes

1. ^ Physics notes - Kinetic energy in the CM frame. Duke.edu. Accessed 2007-11-24.
2. ^ In Einstein's original Über die spezielle und die allgemeine Relativitätstheorie (Zu Seite 41) and in most translations (e.g. Relativity - The Special and General Theory) kinetic energy is defined as mc2 / sqrt(1 − v2 / c2).

## References

• Serway, Raymond A.; Jewett, John W. (2004). Physics for Scientists and Engineers, 6th ed., Brooks/Cole. ISBN 0-534-40842-7.
• Tipler, Paul (2004). Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics, 5th ed., W. H. Freeman. ISBN 0-7167-0809-4.
• Tipler, Paul; Llewellyn, Ralph (2002). Modern Physics, 4th ed., W. H. Freeman. ISBN 0-7167-4345-0.
• School of Mathematics and Statistics, University of St Andrews (2000). Biography of Gaspard-Gustave de Coriolis (1792-1843). Retrieved on 2006-03-03.
• Oxford Dictionary, Oxford Dictionary 1998
Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 62nd day of the year (63rd in leap years) in the Gregorian calendar. ...

Results from FactBites:

 Kinetic energy - Wikipedia, the free encyclopedia (1137 words) Kinetic energy (SI unit: the Joule) is energy that a body possesses as a result of its motion. In another frame of reference the additional kinetic energy is that corresponding to the total mass and the speed of the center of mass. Relativity theory states that the kinetic energy of an object grows towards infinity as its speed approaches the speed of light, and thus that it is impossible to accelerate an object to this boundary.
 Kinetic energy - definition of Kinetic energy in Encyclopedia (485 words) The kinetic energy of a body is equal to the amount of work needed to establish its velocity and rotation, starting from rest. Relativity theory states that the kinetic energy of an object grows towards infinity as its velocity approaches the speed of light, and thus that it is impossible to accelerate an object to this boundary. The relationship between heat, temperature and kinetic energy of atoms and molecules is the subject of statistical mechanics.
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