Kinetic
energy

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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.
The sum of kinetic and potential energy in the system remains constant,
ignoring losses to friction.

Common symbol(s):

KE, Ek, or T

SI unit:

joule
(J)

Derivations from other quantities:

Ek = ½mv2

Ek =
Et+Er

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v

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In physics,
the kinetic energy of an object is the energy which it possesses due to
its motion.[1]
It is defined as the work needed to accelerate
a body of a given mass from rest to its stated velocity. Having gained this
energy during its acceleration, the body
maintains this kinetic energy unless its speed changes. The same amount of work
is done by the body in decelerating from its current speed to a state of
rest.

In classical mechanics,
the kinetic energy of a non-rotating object of mass m traveling at a speed v is ½
mv². In relativistic
mechanics, this is only a good approximation when v is much less than
the speed of light.

Contents

1 History and
etymology

2 Introduction

3 Newtonian kinetic
energy

3.1 Kinetic energy of rigid
bodies

3.1.1 Derivation

3.2 Rotating
bodies

3.3 Kinetic energy of
systems

3.4 Frame of
reference

3.5 Rotation in
systems

4 Relativistic kinetic energy
of rigid bodies

4.1 General
relativity

5 Kinetic energy in quantum
mechanics

6 See also

7 Notes

8 References

History and
etymology[edit]

The adjective kinetic has its roots in the Greek word κίνησις
(kinesis)
meaning motion. The dichotomy between kinetic energy and potential energy can be
traced back to Aristotle's concepts of actuality and
potentiality.[citation
needed]

The principle in classical mechanics
that E ∝ mv² was first developed by Gottfried Leibniz and
Johann Bernoulli, who
described kinetic energy as the living force, vis viva. Willem 's
Gravesande of the Netherlands provided experimental evidence of this
relationship. By dropping weights from different heights into a block of clay,
Willem 's
Gravesande determined that their penetration depth was proportional to the
square of their impact speed. Émilie du
Châtelet recognized the implications of the experiment and published an
explanation.[2]

The terms kinetic energy and work in 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. William
Thomson, later Lord Kelvin, is given the credit for coining the term
"kinetic energy" c. 1849–51.[3][4]

Introduction[edit]

Energy occurs
in many forms, including chemical energy, thermal energy, electromagnetic
radiation, gravitational
energy, electric energy, elastic energy, nuclear energy,
and rest energy. These can be
categorized in two main classes: potential energy and
kinetic energy.

Kinetic energy may be best understood by examples that demonstrate how it is
transformed to and from other forms of energy. For example, a cyclist uses chemical
energy provided by food to accelerate a bicycle to a chosen speed. On a
level surface, this speed can be maintained without further work, except to
overcome air resistance and friction. The
chemical energy has been converted into kinetic energy, the energy of motion,
but the process is not completely efficient and produces heat within the
cyclist.

The kinetic energy in the moving cyclist and the bicycle 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 never regains all of its speed without
additional pedaling. 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 generate some electrical energy on the descent. The bicycle would
be traveling slower at the bottom of the hill than without the generator because
some of the energy has been diverted into electrical energy. Another possibility
would be for the cyclist to apply the brakes, in which case the kinetic energy
would be dissipated through friction as heat.

Like any physical quantity which is a function of velocity, the kinetic
energy of an object depends on the relationship between the object and the
observer's frame of reference.
Thus, the kinetic energy of an object is not invariant.

Spacecraft use chemical
energy to launch and gain considerable kinetic energy to reach orbital
velocity. In a perfectly circular orbit, this kinetic energy remains
constant because there is almost no friction in near-earth space. However it
becomes apparent at re-entry when some of the kinetic energy is converted to
heat. If the orbit is elliptical or hyperbolic, then
throughout the orbit kinetic and potential energy are
exchanged; kinetic energy is greatest and potential energy lowest at closest
approach to the earth or other massive body, while potential energy is greatest
and kinetic energy the lowest at maximum distance. Without loss or gain,
however, the sum of the kinetic and potential energy remains constant.

Kinetic energy can be passed from one object to another. In the game of billiards, the player imposes
kinetic energy on the cue ball by striking it with the cue stick. If the cue
ball collides with another ball, it slows down dramatically and the ball it
collided with accelerates to a speed as the kinetic energy is passed on to it.
Collisions in billiards are
effectively elastic collisions, in
which kinetic energy is preserved. In inelastic
collisions, kinetic energy is dissipated in various forms of energy, such as
heat, sound, binding energy (breaking bound structures).

Flywheels
have been developed as a method of energy storage.
This illustrates that kinetic energy is also stored in rotational motion.

Several mathematical descriptions of kinetic energy exist that describe it in
the appropriate physical situation. For objects and processes in common human
experience, the formula ½mv² given by Newtonian (classical)
mechanics is suitable. However, if the speed of the object is comparable to
the speed of light, relativistic effects
become significant and the relativistic formula is used. If the object is on the
atomic or sub-atomic scale, quantum mechanical
effects are significant and a quantum mechanical model must be employed.

Newtonian kinetic
energy[edit]

Kinetic energy of
rigid bodies[edit]

In classical mechanics,
the kinetic energy of a point object (an object so small that its mass
can be assumed to exist at one point), or a non-rotating rigid body depends on the mass of the body as well
as its speed. The
kinetic energy is equal to the mass multiplied by the square of the speed,
multiplied by the constant 1/2. In formula form:

where
is the mass and
is the speed (or the velocity) of the body. In SI units (used for most modern
scientific work), mass is measured in kilograms, speed in metres per second, and
the resulting kinetic energy is in joules.

For example, one would calculate the kinetic energy of an 80 kg mass
(about 180 lbs) traveling at 18 metres per second (about 40 mph, or
65 km/h) as

When you throw a ball, you do work on it to give it
speed as it leaves your hand. The moving ball can then hit something and push
it, doing work on what it hits. The kinetic energy of a moving object is equal
to the work required to bring it from rest to that speed, or the work the object
can do while being brought to rest: net force × displacement = kinetic
energy, i.e.,

Since the kinetic energy increases with the square of the speed, an object
doubling its speed has four times as much kinetic energy. For example, a car
traveling twice as fast as another requires four times as much distance to stop,
assuming a constant braking force. As a consequence of this quadrupling, it
takes four times the work to double the speed.

The kinetic energy of an object is related to its momentum by the equation:

where:

is momentum

is mass of the body

For the translational kinetic energy, that is the kinetic energy
associated with rectilinear motion, of a rigid body with constant mass ,
whose center of mass is moving
in a straight line with speed ,
as seen above is equal to

where:

is the mass of the body

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

The kinetic energy of any entity depends on the reference frame in which it
is measured. However the total energy of an isolated system, i.e. one which
energy can neither enter nor leave, does not change in whatever reference frame
it is measured. Thus, the chemical energy converted to kinetic energy by a
rocket engine is divided differently between the rocket ship and its exhaust
stream depending upon the chosen reference frame. This is called the Oberth
effect. But the total energy of the system, including kinetic energy, fuel
chemical energy, heat, etc., is conserved over time, regardless of the choice of
reference frame. Different observers moving with different reference frames
disagree on the value of this conserved energy.

The kinetic energy of such systems depends on the choice of reference frame:
the reference frame that gives the minimum value of that energy is the center of momentum
frame, i.e. the reference frame in which the total momentum of the system is
zero. This minimum kinetic energy contributes to the invariant mass of the
system as a whole.

Derivation[edit]

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

where we have assumed the relationship
p = m v. (However, also see the special
relativistic derivation below.)

Applying the product rule we see
that:

Therefore (assuming constant mass so that dm=0), the following can be
seen:

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:

This equation states that the kinetic energy (Ek) is equal
to the integral of the dot
product of the velocity (v) of a body
and the infinitesimal change of
the body's momentum (p). It is
assumed that the body starts with no kinetic energy when it is at rest
(motionless).

Rotating bodies[edit]

If a rigid body is rotating about any line through the center of mass then it
has rotational kinetic
energy ()
which is simply the sum of the kinetic energies of its moving parts, and is thus
given by:

where:

ω is the body's angular velocity

r is the distance of any mass dm from that line

is the body's moment of inertia,
equal to .

(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).

Kinetic energy of
systems[edit]

A system of bodies may have internal kinetic energy due to the relative
motion of the bodies in the system. For example, in the Solar System the planets
and planetoids are orbiting the Sun. In a tank of gas, the molecules are moving
in all directions. The kinetic energy of the system is the sum of the kinetic
energies of the bodies it contains.

A macroscopic body that is stationary (i.e. a reference frame has been chosen
to correspond to the body's center of momentum)
may have various kinds of internal energy at the
molecular or atomic level, which may be regarded as kinetic energy, due to
molecular translation, rotation, and vibration, electron translation and spin,
and nuclear spin. These all contribute to the body's mass, as provided by the
special theory of relativity. When discussing movements of a macroscopic body,
the kinetic energy referred to is usually that of the macroscopic movement only.
However all internal energies of all types contribute to body's mass, inertia,
and total energy.

Frame of reference[edit]

The speed, and thus the kinetic energy of a single object is frame-dependent
(relative): it can take any non-negative value, by choosing a suitable inertial frame
of reference. For example, a bullet passing an observer has kinetic energy
in the reference frame of this observer. The same bullet is stationary from the
point of view of an observer moving with the same velocity as the bullet, and so
has zero kinetic energy.[5]
By contrast, the total kinetic energy of a system of objects cannot be reduced
to zero by a suitable choice of the inertial reference frame, unless all the
objects have the same velocity. In any other case the total kinetic energy has a
non-zero minimum, as no inertial reference frame can be chosen in which all the
objects are stationary. This minimum kinetic energy contributes to the system's
invariant mass, which is
independent of the reference frame.

The total kinetic energy of a system depends on the inertial frame
of reference: it is the sum of the total kinetic energy in a center of momentum
frame and the kinetic energy the total mass would have if it were
concentrated in the center of mass.

This may be simply shown: let
be the relative velocity of the center of mass frame i in the frame
k. Since ,

However, let
the kinetic energy in the center of mass frame,
would be simply the total momentum which is by definition zero in the center of
mass frame, and let the total mass: .
Substituting, we get:[6]

Thus the kinetic energy of a system is lowest with respect to center of
momentum reference frames, i.e., frames of reference in which the center of mass
is stationary (either the center of mass
frame or any other center of momentum
frame). In any other frame of reference there is additional kinetic energy
corresponding to the total mass moving at the speed of the center of mass. The
kinetic energy of the system in the center of momentum
frame is a quantity which is both invariant (all observers see it to be the
same) and is conserved (in an isolated system, it cannot change value, no matter
what happens inside the system).

Rotation in
systems[edit]

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):

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[edit]

See also: Mass in special
relativity and Tests
of relativistic energy and momentum

In special relativity,
we must change the expression for linear momentum.

Using m for rest mass, v and
v for the object's velocity and speed respectively, and c for the
speed of light in vacuum, we assume for linear momentum that ,
where .

Integrating by
parts gives

Remembering that ,
we get:

where E0 serves as an integration constant. Thus:

The constant of integration E0 is found by observing that,
when
and ,
giving

and giving the usual formula:

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 developed by Albert Einstein) to
calculate its kinetic energy.

For a relativistic object the momentum p is equal to:

.

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

.

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:

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
the reciprocal square root and keeping first two terms we get:

,

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:

.

For example, for a speed of 10 km/s (22,000 mph) the correction to
the Newtonian kinetic energy is 0.0417 J/kg (on a Newtonian kinetic energy
of 50 MJ/kg) and for a speed of 100 km/s it is 417 J/kg (on a
Newtonian kinetic energy of 5 GJ/kg), etc.

For higher speeds, the formula for the relativistic kinetic energy[7]
is derived by simply subtracting the rest mass energy from the total energy:

.

The relation between kinetic energy and momentum is more complicated in
this case, and is given by the equation:

.

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

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.

General relativity[edit]

See also: Schwarzschild
geodesics

Using the convention that

where the four-velocity of a
particle is

and
is the proper time of the particle,
there is also an expression for the kinetic energy of the particle in general
relativity.

If the particle has momentum

as it passes by an observer with four-velocity uobs, then
the expression for total energy of the particle as observed (measured in a local
inertial frame) is

and the kinetic energy can be expressed as the total energy minus the rest
energy:

Consider the case of a metric which is diagonal and spatially isotropic
(gtt,gss,gss,gss).
Since

where vα is the ordinary velocity measured w.r.t. the
coordinate system, we get

Solving for ut gives

Thus for a stationary observer (v= 0)

and thus the kinetic energy takes the form

Factoring out the rest energy gives:

This expression reduces to the special relativistic case for the flat-space
metric where

In the Newtonian approximation to general relativity

where Φ is the Newtonian gravitational
potential. This means clocks run slower and measuring rods are shorter near
massive bodies.

Kinetic
energy in quantum mechanics[edit]

Further information: Hamiltonian
(quantum mechanics)

In quantum mechanics,
observables like kinetic energy are represented as operators. For one particle
of mass m, the kinetic energy operator appears as a term in the Hamiltonian
and is defined in terms of the more fundamental momentum operator
as

Notice that this can be obtained by replacing
by
in the classical expression for kinetic energy in terms of momentum,

In the Schrödinger
picture,
takes the form
where the derivative is taken with respect to position coordinates and hence

The expectation value of the electron kinetic energy, ,
for a system of N electrons described by the wavefunction
is a sum of 1-electron operator expectation values:

where
is the mass of the electron and
is the Laplacian operator acting upon
the coordinates of the ith electron and the summation runs
over all electrons.

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 ,
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

where
is known as the von
Weizsäcker kinetic energy functional.

See also[edit]

Energy
portal

Escape velocity

Joule

KE-Munitions

Kinetic
energy per unit mass of projectiles

Kinetic
projectile

Parallel axis
theorem

Potential energy

Recoil

Notes[edit]

^
Jain, Mahesh C. (2009).
Textbook of Engineering Physics (Part I). PHI Learning
Pvt. Ltd. p. 9. ISBN 81-203-3862-6. , Chapter 1, p. 9

^
Judith P. Zinsser
(2007). Emilie du Chatelet: Daring Genius of the Enlightenment.
Penguin. ISBN 0-14-311268-6.

^
Crosbie Smith, M.
Norton Wise. Energy and Empire: A Biographical Study of Lord Kelvin.
Cambridge University Press. p. 866. ISBN 0-521-26173-2.

^
John Theodore Merz
(1912). A History of European Thought in the Nineteenth Century.
Blackwood. p. 139. ISBN 0-8446-2579-5.

^
Sears, Francis Weston;
Brehme, Robert W. (1968). Introduction to the theory of relativity.
Addison-Wesley. p. 127. , Snippet view of page 127

^
Physics notes - Kinetic energy in the CM frame. Duke.edu. Accessed
2007-11-24.

^
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 .

References[edit]

kinetic energy—What it is and how it works.

Oxford Dictionary
1998

School of Mathematics and Statistics,
University of St Andrews (2000). "Biography of Gaspard-Gustave de Coriolis (1792-1843)". Retrieved 2006-03-03.

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.

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