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.

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.

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.^[2] 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.

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.

History and etymology[link]

The adjective kinetic has its roots in the Greek word κίνησις (kinesis) meaning motion.

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.^[3]

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.^[4][5]

Introduction[link]

Energy occurs in many forms, including chemical energy, thermal energy, electromagnetic radiation, gravitational energy, electric energy, elastic energy, nuclear energy, 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 description 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[link]

Kinetic energy of rigid bodies[link]

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, is given by the equation

Failed to parse (Missing texvc executable; please see math/README to configure.): E_k =\tfrac{1}{2} mv^2

where Failed to parse (Missing texvc executable; please see math/README to configure.): m

is the mass and Failed to parse (Missing texvc executable; please see math/README to configure.): v
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

E_k = (1/2) · 80 · 18² J = 12.96 kJ

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.

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

Failed to parse (Missing texvc executable; please see math/README to configure.): E_k = \frac{p^2}{2m}

where:

Failed to parse (Missing texvc executable; please see math/README to configure.): p\;

is momentum

Failed to parse (Missing texvc executable; please see math/README to configure.): m\;

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 Failed to parse (Missing texvc executable; please see math/README to configure.): m\; , whose center of mass is moving in a straight line with speed Failed to parse (Missing texvc executable; please see math/README to configure.): v\; , as seen above is equal to

Failed to parse (Missing texvc executable; please see math/README to configure.): E_t =\tfrac{1}{2} mv^2

where:

Failed to parse (Missing texvc executable; please see math/README to configure.): m\;

is the mass of the body

Failed to parse (Missing texvc executable; please see math/README to configure.): v\;

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[link]

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \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})\,,

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

Applying the product rule we see that:

Failed to parse (Missing texvc executable; please see math/README to configure.): d(\mathbf{v} \cdot \mathbf{v}) = (d \mathbf{v}) \cdot \mathbf{v} + \mathbf{v} \cdot (d \mathbf{v}) = 2(\mathbf{v} \cdot d\mathbf{v}).

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \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:

Failed to parse (Missing texvc executable; please see math/README to configure.): E_k = \int \mathbf{F} \cdot d \mathbf{x} = \int \mathbf{v} \cdot d (m \mathbf{v}) = \int d \left(\frac{m v^2}{2}\right) = \frac{m v^2}{2}.

This equation states that the kinetic energy (E_k) 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[link]

If a rigid body is rotating about any line through the center of mass then it has rotational kinetic energy (Failed to parse (Missing texvc executable; please see math/README to configure.): E_r\, ) which is simply the sum of the kinetic energies of its moving parts, and is thus given by:

Failed to parse (Missing texvc executable; please see math/README to configure.): 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
• Failed to parse (Missing texvc executable; please see math/README to configure.): I\,

is the body's moment of inertia, equal to Failed to parse (Missing texvc executable; please see math/README to configure.): \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).

Kinetic energy of systems[link]

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[link]

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 V be the relative speed of the frame k from the center of mass frame i :

Failed to parse (Missing texvc executable; please see math/README to configure.): E_k = \int \frac{v^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 Failed to parse (Missing texvc executable; please see math/README to configure.): \int \frac{v_i^2 dm}{2} = E_i

the kinetic energy in the center of  mass frame, Failed to parse (Missing texvc executable; please see math/README to configure.):  \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: Failed to parse (Missing texvc executable; please see math/README to configure.):  \int dm = M 

. Substituting, we get:^[6]

Failed to parse (Missing texvc executable; please see math/README to configure.): E_k = E_i + \frac{M V^2}{2}.

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[link]

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):

Failed to parse (Missing texvc executable; please see math/README to configure.): E_k = E_t + E_r \,

where:

E_k is the total kinetic energy

E_t is the translational kinetic energy

E_r 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[link]

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 Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{p}=m\gamma \mathbf{v} , where Failed to parse (Missing texvc executable; please see math/README to configure.): \gamma = 1/\sqrt{1-v^2/c^2} .

Integrating by parts gives

Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \gamma = (1 - v^2/c^2)^{-1/2}\! , we get:

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} 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} - E_0 \end{align}

where E₀ serves as an integration constant. Thus:

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} E_k &= m \gamma (v^2 + c^2 (1 - v^2/c^2)) - E_0 \\ &= m \gamma (v^2 + c^2 - v^2) - E_0 \\ &= m \gamma c^2 - E_0 \end{align}

The constant of integration E₀ is found by observing that, when Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{v }= 0 , \ \gamma = 1\!

and Failed to parse (Missing texvc executable; please see math/README to configure.):  E_k = 0 \!

, giving

Failed to parse (Missing texvc executable; please see math/README to configure.): E_0 = m c^2 \,

and giving the usual formula:

Failed to parse (Missing texvc executable; please see math/README to configure.): 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 developed by Albert Einstein) to calculate its kinetic energy.

For a relativistic object the momentum p is equal to:

Failed to parse (Missing texvc executable; please see math/README to configure.): p = \frac{m v}{\sqrt{1 - (v/c)^2}}

.

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

Failed to parse (Missing texvc executable; please see math/README to configure.): 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:

Failed to parse (Missing texvc executable; please see math/README to configure.): E_\text{rest} = E_0 = 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 the reciprocal square root and keeping first two terms we get:

Failed to parse (Missing texvc executable; please see math/README to configure.): E_k \approx m c^2 \left(1 + \frac{1}{2} v^2/c^2\right) - 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:

Failed to parse (Missing texvc executable; please see math/README to configure.): E_k \approx m c^2 \left(1 + \frac{1}{2} v^2/c^2 + \frac{3}{8} v^4/c^4\right) - 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 (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:

Failed to parse (Missing texvc executable; please see math/README to configure.): E_k = m \gamma c^2 - m c^2 = m c^2\left(\frac{1}{\sqrt{1 - (v/c)^2}} - 1\right)

.

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

Failed to parse (Missing texvc executable; please see math/README to configure.): 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.

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[link]

Using the convention that

Failed to parse (Missing texvc executable; please see math/README to configure.): g_{\alpha \beta} \, u^{\alpha} \, u^{\beta} \, = \, - c^2

where the four-velocity of a particle is

Failed to parse (Missing texvc executable; please see math/README to configure.): u^{\alpha} \, = \, \frac{d x^{\alpha}}{d \tau}

and Failed to parse (Missing texvc executable; please see math/README to configure.): \tau \,

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

Failed to parse (Missing texvc executable; please see math/README to configure.): p_{\beta} \, = \, m \, g_{\beta \alpha} \, u^{\alpha}

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

Failed to parse (Missing texvc executable; please see math/README to configure.): E \, = \, - \, p_{\beta} \, u_{\text{obs}}^{\beta}

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

Failed to parse (Missing texvc executable; please see math/README to configure.): E_{k} \, = \, - \, p_{\beta} \, u_{\text{obs}}^{\beta} \, - \, m \, c^2 \, .

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

Failed to parse (Missing texvc executable; please see math/README to configure.): u^{\alpha} = \frac{d x^{\alpha}}{d t} \frac{d t}{d \tau} = v^{\alpha} u^{t} \,

where v^α is the ordinary velocity measured w.r.t. the coordinate system, we get

Failed to parse (Missing texvc executable; please see math/README to configure.): -c^2 = g_{\alpha \beta} u^{\alpha} u^{\beta} = g_{t t} (u^{t})^2 + g_{s s} v^2 (u^{t})^2 \,.

Solving for u^t gives

Failed to parse (Missing texvc executable; please see math/README to configure.): u^{t} = c \sqrt{\frac{-1}{g_{t t} + g_{s s} v^2}} \,.

Thus for a stationary observer (v= 0)

Failed to parse (Missing texvc executable; please see math/README to configure.): u_{\text{obs}}^{t} = c \sqrt{\frac{-1}{g_{t t}}} \,

and thus the kinetic energy takes the form

Failed to parse (Missing texvc executable; please see math/README to configure.): E_k = - m g_{tt} u^t u_{\text{obs}}^t - m c^2 = m c^2 \sqrt{\frac{g_{tt}}{g_{tt} + g_{ss} v^2}} - m c^2\,.

Factoring out the rest energy gives:

Failed to parse (Missing texvc executable; please see math/README to configure.): E_k = m c^2 \left( \sqrt{\frac{g_{tt}}{g_{tt} + g_{ss} v^2}} - 1 \right) \,.

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

Failed to parse (Missing texvc executable; please see math/README to configure.): g_{t t} = -c^2 \,

Failed to parse (Missing texvc executable; please see math/README to configure.): g_{s s} = 1 \,.

In the Newtonian approximation to general relativity

Failed to parse (Missing texvc executable; please see math/README to configure.): g_{t t} = - \left( c^2 + 2 \Phi \right) \,

Failed to parse (Missing texvc executable; please see math/README to configure.): g_{s s} = 1 - \frac{2 \Phi}{c^2} \,

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[link]

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 Failed to parse (Missing texvc executable; please see math/README to configure.): \hat p

as

Failed to parse (Missing texvc executable; please see math/README to configure.): \hat T = \frac{\hat p^2}{2m}.

Notice that this can be obtained by replacing Failed to parse (Missing texvc executable; please see math/README to configure.): p

by Failed to parse (Missing texvc executable; please see math/README to configure.): \hat p
in the classical expression for kinetic energy in terms of momentum,

Failed to parse (Missing texvc executable; please see math/README to configure.): E_k = \frac{p^2}{2m}.

In the Schrödinger picture, Failed to parse (Missing texvc executable; please see math/README to configure.): \hat p

takes the form Failed to parse (Missing texvc executable; please see math/README to configure.): -i\hbar\nabla 
where the derivative is taken with respect to position coordinates and hence

Failed to parse (Missing texvc executable; please see math/README to configure.): \hat T = -\frac{\hbar^2}{2m}\nabla^2.

The expectation value of the electron kinetic energy, Failed to parse (Missing texvc executable; please see math/README to configure.): \langle\hat{T}\rangle , for a system of N electrons described by the wavefunction Failed to parse (Missing texvc executable; please see math/README to configure.): \vert\psi\rangle

is a sum of 1-electron operator expectation values:

Failed to parse (Missing texvc executable; please see math/README to configure.): \langle\hat{T}\rangle = \bigg\langle\psi \bigg\vert \sum_{i=1}^N \frac{-\hbar^2}{2 m_e} \nabla^2_i \bigg\vert \psi \bigg\rangle = -\frac{\hbar^2}{2 m_e} \sum_{i=1}^N \bigg\langle\psi \bigg\vert \nabla^2_i \bigg\vert \psi \bigg\rangle

where Failed to parse (Missing texvc executable; please see math/README to configure.): m_e

is the mass of the electron and Failed to parse (Missing texvc executable; please see math/README to configure.): \nabla^2_i
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 Failed to parse (Missing texvc executable; please see math/README to configure.): \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

Failed to parse (Missing texvc executable; please see math/README to configure.): T[\rho] = \frac{1}{8} \int \frac{ \nabla \rho(\mathbf{r}) \cdot \nabla \rho(\mathbf{r}) }{ \rho(\mathbf{r}) } d^3r

where Failed to parse (Missing texvc executable; please see math/README to configure.): T[\rho]

is known as the von Weizsäcker kinetic energy functional.

See also[link]

Energy portal

• Potential energy
• Escape velocity
• Joule
• KE-Munitions
• Kinetic energy per unit mass of projectiles
• Kinetic projectile
• Parallel axis theorem
• Recoil

Notes[link]

References[link]

• 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)". http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Coriolis.html. 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. 

Bill Nye
Nye at Bridgewater State University in 2007
Born	(1955-11-27) November 27, 1955 (age 56)[1] Washington, D.C.[1]
Residence	Los Angeles
Fields	Mechanical engineering
Institutions	Boeing Cornell University Planetary Society
Alma mater	Cornell University (BS)
Known for	Bill Nye the Science Guy
Signature

William Sanford "Bill" Nye (born November 27, 1955),^[2] popularly known as Bill Nye the Science Guy, is an American science educator, comedian, television host, actor, mechanical engineer, and scientist. He is best known as the host of the Disney/PBS children's science show Bill Nye the Science Guy (1993–1998) and for his many subsequent appearances in popular media as a science educator.

Early life and career[link]

William Sanford Nye was born in Washington, D.C., the son of Jacqueline (née Jenkins; c. 1920–2000), a codebreaker during World War II, and Edwin Darby "Ned" Nye (died 1997), also a World War II veteran whose experience in a Japanese prisoner of war camp led him to become a sundial enthusiast.^[3][4][5] Nye is a fourth-generation Washington, D.C. resident on his father's side of the family. After attending Lafayette Elementary and Alice Deal Junior High in the city, he was accepted to the private Sidwell Friends School on a partial scholarship, graduating in 1973.^[6][7] He studied mechanical engineering at Cornell University, where one of his professors was Carl Sagan,^[8] and graduated with a Bachelor of Science in 1977.^[9] He was awarded an honorary doctorate by The Johns Hopkins University in May 2008. In May 2011, Nye was awarded an Honorary Doctor of Science degree from Willamette University where he was the keynote speaker for that year's commencement exercises.^[10]

Nye began his career in Seattle at Boeing, where, among other things, he starred in training films and developed a hydraulic pressure resonance suppressor still used in the 747. Later he worked as a consultant in the aeronautics industry. Nye told the St. Petersburg Times in 1999 that he applied to be a NASA astronaut every few years but was always rejected.^[11]

The Science Guy[link]

Nye began his professional entertainment career as an actor on a local sketch comedy television show in Seattle, Almost Live!; Nye attempted to correct the host of Almost Live! after the host pronounced "gigawatt" as "jigowatt."^[12][13] The character name came from the host's comment "Who do you think you are—Bill Nye the Science Guy?" and Nye was thereafter known as such on the program. His other main recurring role on Almost Live! was as Speedwalker, a speedwalking Seattle superhero.

Coincidentally, in 1991–93 he appeared in the live-action educational segments of Back to the Future: The Animated Series with a nonspeaking role as an assistant to Dr. Emmett Brown, played by Christopher Lloyd, in which he would demonstrate science with Lloyd's voice-over. The segments' national popularity led to Nye's hosting the educational television program Bill Nye the Science Guy from 1993 to 1998. Each of the 100 episodes aimed to teach a specific topic in science to a preteen audience, yet it garnered a wide adult audience as well. The show was very popular as a school resource and is still used to this day. He has written several books as The Science Guy. In addition to hosting the show, he was also a writer and producer for the series, which was filmed entirely in Seattle.

When portraying "the Science Guy", Nye wears a light blue lab coat and a bow tie and takes on the persona of an excited, jocular science educator. This popular image of Nye has been parodied by numerous sources, including the webcomic xkcd^[14] and the satirical news organization The Onion.^[15][16] In response to the fake headline "Crack Nearly Killed Me," Nye took the joke in good humor and sent The Onion an email thanking them for "dealing compassionately with this matter."^[17]

He appeared in his Science Guy persona alongside Ellen DeGeneres and Alex Trebek in Ellen's Energy Adventure, an attraction that has played since 1996 at the Universe of Energy pavilion inside Epcot at Walt Disney World. He also has a voice-over at the DINOSAUR attraction in Disney's Animal Kingdom park, where he tells guests about the dinosaurs while they queue for the ride. In addition, he appears in the "Design Lab" of CyberSpace Mountain inside DisneyQuest at Walt Disney World, where he refers to himself as "Bill Nye the Coaster Guy."

[edit] Post-Science Guy career

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Bill Nye "The Science Guy" and Executive Director of The Planetary Society received the Committee for Skeptical Inquiry's "In Praise of Reason" Award at CSICON 2011 in New Orleans

Nye remained interested in science education through entertainment. He created a 13-episode PBS KCTS-TV series about science, called The Eyes of Nye, aimed at an older audience than his previous show had been. Airing in 2005, it often featured episodes based on politically relevant themes such as genetically modified food, global warming, and race.

He played a science teacher in Disney's 1998 TV movie The Principal Takes a Holiday; he made a hovercraft in order to demonstrate science in an unusual classroom manner. From 2000 to 2002, Nye was the technical expert in BattleBots. In 2004 and 2005, Nye hosted 100 Greatest Discoveries, an award-winning series produced by THINKFilm for The Science Channel and in high definition on the Discovery HD Theater. He was also host of an eight-part Discovery Channel series called Greatest Inventions with Bill Nye.

Nye has guest-starred in several episodes of the crime drama Numb3rs as an engineering faculty member. A lecture Nye gave several years ago on exciting children about math was an inspiration for creating the Numb3rs show.^[18]

Bill joined the American Optometric Association in a multimedia advertising campaign to educate parents on the importance of getting their children a comprehensive eye examination.^[19]

He was a regular in TV Land discussions. He has also made guest appearances on the VH1 reality show America's Most Smartest Model.^[20]

Nye appears in segments of The Climate Code on The Weather Channel, telling his personal ways of saving energy. He still makes regular appearances on the show, often asking quiz questions.

As of fall 2008, Nye also appears on the daytime game show Who Wants to Be a Millionaire as part of the show's reintroduced "Ask the Expert" lifeline. In 2008, he also hosted Stuff Happens, a show on the then new Planet Green network.

In November 2008, Nye appeared in an acting role as himself in the fifth-season episode "Brain Storm" of Stargate Atlantis alongside fellow television personality and astrophysicist Neil deGrasse Tyson.^[21]

Nye has appeared numerous times on the talk show Larry King Live, speaking about topics such as global warming^[22] and UFOs.^[23] He argued that global warming is an issue that should be addressed by governments of the world in part because it could be implicated in the record-setting 2005 Atlantic hurricane season. On UFOs he has been skeptical of extraterrestrial explanations for sightings such as those at Roswell and Malmstrom Air Force Base in 1967.

In 2009, portions of Bill Nye's shows were used as lyrics and portions of the second Symphony of Science science education music video by composer John Boswell.

Nye recorded a short YouTube video (as himself, not his TV persona) advocating clean energy climate change legislation on behalf of Al Gore's Repower America campaign in October 2009.^[24]

Nye (as his TV persona) also made a guest appearance on The Dr. Oz Show.

On March 12, 2011, Nye made an appearance on CNN to discuss the evolving nuclear incidents in Japan as a result of the devastating earthquake and tsunami there. Nye erroneously stated that cesium is used to "slow and control" the nuclear reaction. ^[25] In reality, cesium (specifically cesium-137) is a nuclear fission product, not a control rod material. Nye also erroneously stated that the nuclear reactor involved in the Three Mile Island incident is still running and that the use of boron to slow the nuclear chain reaction is uncommon, when in fact boron-10 is commonly used in control rods, and is circulated in the coolant of reactors in the United States, as well as stored on site as a method of emergency shutdown.^[26][27]

A solar noon clock atop Rhodes Hall was gifted from Nye to Cornell on Aug. 27 following a public lecture that filled the 715-seat Statler Auditorium. Nye talked about his father's passion for sundials and timekeeping, his time at Cornell, his work on the sundials mounted on the Mars rovers and the story behind the Bill Nye Solar Noon Clock. ^[28]

Life outside television[link]

Nye speaking in 2010.

In the early 2000s, Nye assisted in the development of a small sundial that was included in the Mars Exploration Rover missions. Known as MarsDial, it included small colored panels to provide a basis for color calibration in addition to helping keep track of time.^[29] From 2005 to 2010 Nye was the vice president of The Planetary Society, an organization that advocates space science research and the exploration of other planets, particularly Mars.^[30] He became the organization's second Executive Director in September 2010 when Louis Friedman stepped down.^[31][32]

In November 2010, Nye became the face of a new permanent exhibition at the Chabot Space & Science Center in Oakland, CA. Bill Nye’s Climate Lab features Nye as commander of the Clean Energy Space Station, and invites visitors on an urgent mission to thwart climate change. Beginning with a view of Planet Earth from space, visitors explore air, water, and land galleries to discover how climate change affects Earth’s connected systems, and how to use the Sun, wind, land, and water to generate clean energy. In an interview about the exhibit, Nye said, “Everything in the exhibit is geared to showing you that the size of the problem of climate change is big. Showing you a lot about energy use... It’s a huge opportunity... We need young people, entrepreneurs, young inventors, young innovators to change the world.” ^[33]

He holds several United States patents,^[34] including one for ballet pointe shoes^[30] and another for an educational magnifying glass created by filling a clear plastic bag with water.^[35][36] From 2001–2006 Nye served as Frank H.T. Rhodes Class of '56 University Professor at Cornell University.^[9][37] Nye supported the 2006 reclassification of Pluto from a planet to a dwarf planet by the International Astronomical Union.^[38]

Since 2006, Nye has lived in Los Angeles in a house with modifications, though has owned a house on Mercer Island.^{[citation needed]} As of July 2007, Nye and environmental activist Ed Begley, Jr. are engaging in a friendly competition "to see who could have the lowest carbon footprint," according to Begley.^[39] In a 2008 interview, Nye joked that he wants to "crush Ed Begley" in their environmental competition.^[40] Nye and Begley are neighbors in Los Angeles, and sometimes dine together at a local vegetarian restaurant.^[40] Nye often appears on Begley's Planet Green reality show Living with Ed.

Nye made an appearance in Palmdale's 2010 video "Here Comes the Summer"^[41]; the band's lead singer Kay Hanley is his neighbor.

Nye is a fellow of the Committee for Skeptical Inquiry, a U.S. non-profit scientific and educational organization whose aim is to promote scientific inquiry, critical investigation, and the use of reason in examining controversial and extraordinary claims.^[42]

Interviewed by John Rael for the Independent Investigation Group IIG, Nye stated that his "concern right now... scientific illiteracy... you don't [the public] have enough rudimentary knowledge of the universe to evaluate claims."^[43]

Personal life[link]

Nye announced his engagement during an appearance on The Late Late Show with Craig Ferguson and was married briefly to his fiancée of five months, author Blair Tindall, on February 3, 2006. The ceremony was performed by Rick Warren at The Entertainment Gathering at the Skirball Cultural Center in Los Angeles. Yo-Yo Ma provided the music.^[44] Nye left the relationship seven weeks later when the marriage license was declared invalid;^[45] their impromptu ceremony, which preceded the license purchase, violated California state law, said Tindall in a radio interview.^[46] Nye later secured a temporary restraining order against Tindall after she entered his property late at night to pour weed killer in his garden.^[45]

Nye enjoys baseball and occasionally does experiments involving the physics of the game. As a longtime Seattle resident before becoming an entertainer, he is said to have been a fan of the Seattle Mariners, although recently he has voiced his preference (as a D.C. native) for the Washington Nationals.^[6]

References[link]

External links[link]

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Biography portal

Wikimedia Commons has media related to: Bill Nye

• Official website
• Bill Nye at the Internet Movie Database
• Bill Nye on Twitter
• Nye-Tindall wedding video
• An interview/discussion with Nye as guest on the radio show Loveline
• "Changing The World With Science Education" interview on Point of Inquiry.
• Seattle Times interview (April 26, 2004)
• Science Channel interview