Title page of the 1st edition of Isaac Newton's Principia defining the laws of motion.

In classical mechanics, linear momentum or translational momentum (pl. momenta; SI unit kg•m/s, or, equivalently, N•s) is the product of the mass and velocity of an object:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{p} \equiv m\mathbf{v}.

Like velocity, linear momentum is a vector quantity, possessing a direction as well as a magnitude. Linear momentum is also a conserved quantity, meaning that if a closed system is not affected by external forces, its total linear momentum cannot change. Although originally expressed in Newton's second law, the conservation of linear momentum also holds in special relativity and, with appropriate definitions, a (generalized) linear momentum conservation law holds in electrodynamics, quantum mechanics, quantum field theory, and general relativity. In relativistic mechanics, non-relativistic linear momentum is further multiplied by the Lorentz factor.

History of the concept[link]

Mōmentum was not merely the motion, which was mōtus, but was the power residing in a moving object, captured by today's mathematical definitions. A mōtus, "movement", was a stage in any sort of change,^[1] while velocitas, "swiftness", captured only speed. The concept of momentum in classical mechanics was originated by a number of great thinkers and experimentalists. The first of these was Byzantine philosopher John Philoponus, in his commentary to Aristotle´s Physics. As regards the natural motion of bodies falling through a medium, Aristotle's verdict that the speed is proportional to the weight of the moving bodies and indirectly proportional to the density of the medium is disproved by Philoponus through appeal to the same kind of experiment that Galileo was to carry out centuries later.^[2] This idea was refined by the European philosophers Peter Olivi and Jean Buridan. Buridan referred to impetus being proportional to the weight times the speed.^[3][4] Moreover, Buridan's theory was different to his predecessor's in that he did not consider impetus to be self dissipating, asserting that a body would be arrested by the forces of air resistance and gravity which might be opposing its impetus.^[5]

René Descartes believed that the total "quantity of motion" in the universe is conserved, where the quantity of motion is understood as the product of size and speed. This should not be read as a statement of the modern law of momentum, since he had no concept of mass as distinct from weight and size, and more importantly he believed that it is speed rather than velocity that is conserved. So for Descartes if a moving object were to bounce off a surface, changing its direction but not its speed, there would be no change in its quantity of motion.^[6] Galileo, later, in his Two New Sciences, used the Italian word "impeto".

The extent to which Isaac Newton contributed to the concept has been much debated. The answer is apparently nothing, except to state more fully and with better mathematics what was already known. Yet for scientists, this was the death knell for Aristotelian physics and supported other progressive scientific theories (i.e., Kepler's laws of planetary motion). Conceptually, the first and second of Newton's Laws of Motion had already been stated by John Wallis in his 1670 work, Mechanica sive De Motu, Tractatus Geometricus: "the initial state of the body, either of rest or of motion, will persist" and "If the force is greater than the resistance, motion will result".^[7] Wallis uses momentum and vis for force. Newton's Philosophiæ Naturalis Principia Mathematica, when it was first published in 1687, showed a similar casting around for words to use for the mathematical momentum. His Definition II^[8] defines quantitas motus, "quantity of motion", as "arising from the velocity and quantity of matter conjointly", which identifies it as momentum.^[9] Thus when in Law II he refers to mutatio motus, "change of motion", being proportional to the force impressed, he is generally taken to mean momentum and not motion.^[10] It remained only to assign a standard term to the quantity of motion. The first use of "momentum" in its proper mathematical sense is not clear but by the time of Jenning's Miscellanea in 1721, four years before the final edition of Newton's Principia Mathematica, momentum M or "quantity of motion" was being defined for students as "a rectangle", the product of Q and V, where Q is "quantity of material" and V is "velocity", s/t.^[11]

The Oxford English Dictionary dates the first print use of the word to 1699: John Keill, Reflections upon the Theory of the earth [written by himself] occasion'd by a late Examination of it, a letter (London).^[12]

Some languages, such as French and Italian still lack a single term for momentum, and use a phrase such as the literal translation of "quantity of motion". In Bulgarian and in Dutch the linear momentum is typically referred to as impulse, while the angular momentum is called momentum and the impulse has no distinct name.

Linear momentum of a particle[link]

Newton's apple in Einstein's elevator. In person A's frame of reference, the apple has non-zero velocity and momentum. In the elevator's and person B's frames of reference, it has zero velocity and momentum.

If an object is moving in any reference frame, then it has momentum in that frame. It is important to note that momentum is frame dependent. That is, the same object may have a certain momentum in one frame of reference, but a different amount in another frame. For example, a moving object has momentum in a reference frame fixed to a spot on the ground, while at the same time having 0 momentum in a reference frame attached to the object's center of mass.

The amount of momentum that an object has depends on two physical quantities: the mass and the velocity of the moving object in the frame of reference. In physics, the usual symbol for momentum is a boldface p (bold because it is a vector); so this can be written

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{p}= m \mathbf{v}\,,

where p is the momentum, m is the mass and v is the velocity.

Example: a model airplane of 1 kg traveling due north at 1 m/s in straight and level flight has a momentum of 1 kg•m/s due north measured from the ground. To the dummy pilot in the cockpit it has a velocity and momentum of zero.

According to Newton's second law, the rate of change of the momentum of a particle is proportional to the resultant force acting on the particle and is in the direction of that force. The derivation of force from momentum is given below.

Failed to parse (Missing texvc executable; please see math/README to configure.): \sum{\mathbf{F}} = {d\mathbf{p} \over dt} = m{d\mathbf{v} \over dt} + \mathbf{v}{dm \over dt}

Given that mass is constant, the second term of the derivative is zero (Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle \mathbf{v}{dm \over dt} = 0 ). We can therefore write the following:

Failed to parse (Missing texvc executable; please see math/README to configure.): \sum{\mathbf{F}} = m\mathbf{a}\,

or just simply

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{F}= m \mathbf{a}\,,

where F is understood to be the net force (or resultant).

Example: a model airplane of 1 kg accelerates from rest to a velocity of 1 m/s due north in 1 s. The thrust required to produce this acceleration is 1 newton. The change in momentum is 1 kg•m/s. To the dummy pilot in the cockpit there is no change of momentum. Its pressing backward in the seat is a reaction to the unbalanced thrust, shortly to be balanced by the drag.

Linear momentum of a system of particles[link]

Relating to mass and velocity[link]

The linear momentum of a system of particles is the vector sum of the momenta of all the individual objects in the system:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{p}= \sum_{i = 1}^n m_i \mathbf{v}_i = m_1 \mathbf{v}_1 + m_2 \mathbf{v}_2 + m_3 \mathbf{v}_3 + \cdots + m_n \mathbf{v}_n\,,

where p is the total momentum of the particle system, m_i and v_i are the respective mass and velocity of the i-th object, and n is the number of objects in the system.

It can be shown that, in the center of mass frame the momentum of a system is zero. Additionally, the momentum in a frame of reference that is moving at a velocity v_cm with respect to that frame is simply:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{p}= m\mathbf{v}_\text{cm}\,,

where:

Failed to parse (Missing texvc executable; please see math/README to configure.): m=\sum_{i = 1}^n m_i\,.

This is known as Euler's first law.^[13][14]

Relating to force – general equations of motion[link]

Motion of a material body

The linear momentum of a system of particles can also be defined as the product of the total mass, m, of the system times the velocity, v_cm, of the center of mass.

Failed to parse (Missing texvc executable; please see math/README to configure.): \sum{\mathbf{F}} = {d\mathbf{p} \over dt}= m \frac{d\mathbf{v}_{cm}}{dt}=m\mathbf{a}_{cm}\,

This is a special case of Newton's second law (if mass is constant).

For a more general derivation using tensors, we consider a moving body (see Figure), assumed as a continuum, occupying a volume V, at a time t, having a surface area S, with defined traction or surface forces per unit area represented by the stress vector Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle T_i^{(n)}\,

acting on every point of every body surface (external and internal), body forces Fi per unit of volume on every point within the volume V, and a velocity field vi, prescribed throughout the body. Following the previous equation, the linear momentum of the system is:

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_S T_i^{(n)}dS + \int_V F_i dV = \frac{d}{dt}\int_V \rho\, v_i\, dV\,.

By definition the stress vector is defined as Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle T_i^{(n)} \equiv \sigma_{ij}n_j\, , then

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_S \sigma_{ij}n_j\, dS + \int_V F_i\, dV = \frac{d}{dt}\int_V \rho\, v_i\, dV\,.

Using the Gauss's divergence theorem to convert a surface integral to a volume integral gives (we denote Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle \partial_j \equiv \frac{\partial}{\partial x_j}\,

as the differential operator):

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_V \partial_j\sigma_{ij}\, dV + \int_V F_i\, dV = \frac{d}{dt}\int_V \rho\,v_i\, dV\,.

Now we only need to take care of the right side of the equation. We have to be careful, since we cannot just take the differential operator under the integral. This is because while the motion of the continuum body is taking place (the body is not necessarily solid), the volume we are integrating on can change with time too. So the above integral will be:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{d}{dt}\int \rho\,v_i\, dV=\int \frac{\partial (\rho v_i)}{\partial t}\, dV +\oint \rho v_i v_k n_k dA\,.

Performing the differentiation in the first part, and applying the divergence theorem on the second part we obtain:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{d}{dt}\int \rho\,v_i\, dV =\int \left[ \left(\rho\frac{\partial v_i}{\partial t}+v_i\frac{\partial \rho}{\partial t}\right)+\partial_k (\rho v_i v_k)\right]\, dV\,.

Now the second term inside the integral is: Failed to parse (Missing texvc executable; please see math/README to configure.): \partial_k (\rho v_i v_k)=\rho v_k \cdot \partial_k v_i +v_i\partial_k(\rho v_k)\,.

Plugging this into the previous equation, and rearranging the terms, we get:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{d}{dt}\int \rho\,v_i\, dV=\int\rho\left[\frac{\partial}{\partial t}+v_k\partial_k\right]v_i\,dV +\int\left[\frac{\partial\rho}{\partial t}+\partial_k(\rho v_k)\right]v_i\,dV\,.

We can easily recognize the two integral terms in the above equation. The first integral contains the convective derivative of the velocity vector, and the second integral contains the change and flow of mass in time. Now lets assume that there are no sinks and sources in the system, that is mass is conserved, so this term is zero. Hence we obtain:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{d}{dt}\int \rho\,v_i\, dV=\int \rho\,\frac{Dv_i}{Dt}\, dV\,

putting this back into the original equation:

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_V \left[ \partial_j\sigma_{ij} + F_i - \rho \frac{D v_i}{Dt}\right]\, dV = 0\,.

For an arbitrary volume the integrand itself must be zero, and we have the Cauchy's equation of motion

Failed to parse (Missing texvc executable; please see math/README to configure.): \partial_j\sigma_{ij} + F_i = \rho \frac{D v_i}{Dt}\,.

As we see the only extra assumption we made is that the system doesn't contain any mass sources or sinks, which means that mass is conserved. So this equation is valid for the motion of any continuum, even for that of fluids. If we are examining elastic continua only then the second term of the convective derivative operator can be neglected, and we are left with the usual time derivative, of the velocity field.

If a system is in equilibrium, the change in momentum with respect to time is equal to 0, as there is no acceleration

Failed to parse (Missing texvc executable; please see math/README to configure.): \sum{\mathbf{F}} = {d\mathbf{p} \over dt}=\ m\mathbf{a}_{cm}= 0\,

or using tensors,

Failed to parse (Missing texvc executable; please see math/README to configure.): \partial_j\sigma_{ij} + F_i = 0\,.

These are the equilibrium equations which are used in solid mechanics for solving problems of linear elasticity. In engineering notation, the equilibrium equations are expressed in Cartesian coordinates as

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial \sigma_x}{\partial x} + \frac{\partial \tau_{yx}}{\partial y} + \frac{\partial \tau_{zx}}{\partial z} + F_x = 0\,

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial \tau_{xy}}{\partial x} + \frac{\partial \sigma_y}{\partial y} + \frac{\partial \tau_{zy}}{\partial z} + F_y = 0\,

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial \tau_{xz}}{\partial x} + \frac{\partial \tau_{yz}}{\partial y} + \frac{\partial \sigma_z}{\partial z} + F_z = 0\,.

Conservation of linear momentum[link]

This example (m1 = 4kg, u1 = 5m/s, m2 = 4kg, u2=0m/s) of collision shows carts in a perfectly inelastic collision—notice the center of mass of the masses m1 and m2 moves at a constant 2.5m/s throughout the motion.

The law of conservation of linear momentum is a fundamental law of nature, and it states that if no external force acts on a closed system of objects, the momentum of the closed system remains constant. One of the consequences of this is that the center of mass of any system of objects will always continue with the same velocity unless acted on by a force from outside the system.

Conservation of momentum is a mathematical consequence of the homogeneity (shift symmetry) of space (position in space is the canonical conjugate quantity to momentum). That is, conservation of momentum is equivalent to the fact that the physical laws do not depend on position.

In analytical mechanics the conservation of momentum is a consequence of translational invariance of Lagrangian in the absence of external forces. It can be proven that the total momentum is a constant of motion by making an infinitesimal translation of Lagrangian and then equating it with non translated Lagrangian. This is a special case of Noether's theorem .^[15]

In an isolated system (one where external forces are absent) the total momentum will be constant: this is implied by Newton's first law of motion. Newton's third law of motion, the law of reciprocal actions, which dictates that the forces acting between systems are equal in magnitude, but opposite in sign, is due to the conservation of momentum.

Since position in space is a vector quantity, momentum (being the canonical conjugate of position) is a vector quantity as well—it has direction. Thus, when a gun is fired, the final total momentum of the system (the gun and the bullet) is the vector sum of the momenta of these two objects. Assuming that the gun and bullet were at rest prior to firing (meaning the initial momentum of the system was zero), the final total momentum must also equal 0.

In an isolated system with only two objects, the change in momentum of one object must be equal and opposite to the change in momentum of the other object. Mathematically,

Failed to parse (Missing texvc executable; please see math/README to configure.): \Delta \mathbf{p}_1 = -\Delta \mathbf{p}_2\,.

Momentum has the special property that, in a closed system, it is always conserved, even in collisions and separations caused by explosive forces. Kinetic energy, on the other hand, is not conserved in collisions if they are inelastic. Since momentum is conserved it can be used to calculate an unknown velocity following a collision or a separation if all the other masses and velocities are known.

A common problem in physics that requires the use of this fact is the collision of two particles. Since momentum is always conserved, the sum of the momenta before the collision must equal the sum of the momenta after the collision:

Failed to parse (Missing texvc executable; please see math/README to configure.): m_1 \mathbf u_{1} + m_2 \mathbf u_{2} = m_1 \mathbf v_{1} + m_2 \mathbf v_{2}\,,

where u₁ and u₂ are the velocities before collision, and v₁ and v₂ are the velocities after collision.

Determining the final velocities from the initial velocities (and vice versa) depend on the type of collision. There are two types of collisions that conserve momentum: elastic collisions, which also conserve kinetic energy, and inelastic collisions, which do not.

Elastic collisions[link]

Example (m1 = 4kg, u1 = 5m/s, m2 = 4kg, u2=0m/s) of collision carts in perfectly elastic collision showing the equations of momentum and kinetic energy both conserved

A collision between two pool balls is a good example of an almost totally elastic collision, due to their high rigidity; a totally elastic collision exists only in theory, occurring between bodies with mathematically infinite rigidity. In addition to momentum being conserved when the two balls collide, the sum of kinetic energy before a collision must equal the sum of kinetic energy after:

Failed to parse (Missing texvc executable; please see math/README to configure.): \tfrac{1}{2} m_1 u_{1}^2 + \tfrac{1}{2} m_2 u_{2}^2 = \tfrac{1}{2} m_1 v_{1}^2 + \tfrac{1}{2} m_2 v_{2}^2\,.

In one dimension[link]

When the initial velocities are known, the final velocities for a head-on collision are given by

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{v}_{1} = \left( \frac{m_1 - m_2}{m_1 + m_2} \right) \mathbf{u}_{1} + \left( \frac{2 m_2}{m_1 + m_2} \right) \mathbf{u}_{2}\,

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{v}_{2} = \left( \frac{m_2 - m_1}{m_1 + m_2} \right) \mathbf{u}_{2} + \left( \frac{2 m_1}{m_1 + m_2} \right) \mathbf{u}_{1}\,.

This example m1=1000kg, u1=5m/s, m2=0.1kg, u2=0m/s, the final velocities are approximately given v1=4.999m/s, v2=9.999m/s

When the first body is much more massive than the other (that is, m₁ » m₂), the final velocities are approximately given by

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{v}_{1} = \mathbf{u}_{1}\,

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{v}_{2} = 2\mathbf{u}_{1} - \mathbf{u}_{2}\,.

Thus the more massive body does not change its velocity, and the less massive body travels at twice the velocity of the more massive body less its own original velocity. Assuming both masses were heading towards each other on impact, the less massive body is now therefore moving in the opposite direction at twice the speed of the more massive body plus its own original speed.

A Newton's cradle demonstrates conservation of momentum.

In a head-on collision between two bodies of equal mass (that is, m₁ = m₂), the final velocities are given by

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{v}_1 = \mathbf{u}_2\,

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{v}_2 = \mathbf{u}_1\,.

Thus the bodies simply exchange velocities. If the first body has nonzero initial velocity u₁ and the second body is at rest, then after collision the first body will be at rest and the second body will travel with velocity u₁. This phenomenon is demonstrated by Newton's cradle.

In multiple dimensions[link]

In the case of objects colliding in more than one dimension, as in oblique collisions, the velocity is resolved into orthogonal components with one component perpendicular to the plane of collision and the other component or components in the plane of collision. The velocity components in the plane of collision remain unchanged, while the velocity perpendicular to the plane of collision is calculated in the same way as the one-dimensional case.

For example, in a two-dimensional collision, the momenta can be resolved into x and y components. We can then calculate each component separately, and combine them to produce a vector result. The magnitude of this vector is the final momentum of the isolated system.

Perfectly inelastic collisions[link]

Example (m1 = 4kg, u1 = 5m/s, m2 = 4kg, u2=0m/s) of collision carts in perfectly inelastic collision showing the equations of momentum conserved and kinetic energy associated with some energy loss.

A common example of a perfectly inelastic collision is when two snowballs collide and then stick together afterwards. This equation describes the conservation of momentum:

Failed to parse (Missing texvc executable; please see math/README to configure.): m_1 \mathbf u_{1} + m_2 \mathbf u_{2} = \left( m_1 + m_2 \right) \mathbf v\,

It can be shown that a perfectly inelastic collision is one in which the maximum amount of kinetic energy is converted into other forms. For instance, if both objects stick together after the collision and move with a final common velocity, one can always find a reference frame in which the objects are brought to rest by the collision and 100% of the kinetic energy is converted. This is true even in the relativistic case and utilized in particle accelerators to efficiently convert kinetic energy into new forms of mass-energy (i.e. to create massive particles).

Coefficient of restitution[link]

The coefficient of restitution is defined as the ratio of relative velocity of separation to relative velocity of approach. It is a ratio hence it is a dimensionless quantity. The coefficient of restitution is given by:

Failed to parse (Missing texvc executable; please see math/README to configure.): C_R = \frac{v_2 - v_1}{u_1 - u_2}

for two colliding objects, where

v₁ is the scalar final velocity of the first object after impact,

v₂ is the scalar final velocity of the second object after impact,

u₁ is the scalar initial velocity of the first object before impact,

u₂ is the scalar initial velocity of the second object before impact.

A perfectly elastic collision implies that C_R is 1. So the relative velocity of approach is same as the relative velocity of separation of the colliding bodies.

Inelastic collisions have (C_R < 1). In case of a perfectly inelastic collision the relative velocity of separation of the centre of masses of the colliding bodies is 0. Hence the bodies stick together after collision.

Explosions[link]

An explosion occurs as a result of a chain reaction that transforms potential energy into kinetic energy displacing the surrounding material. Explosions do not conserve potential energy. Instead, potential energy stored in chemical, mechanical, or nuclear form, is transformed into kinetic energy, acoustic energy, and electromagnetic radiation.

See the inelastic collision page for more details.

Modern definitions of momentum[link]

Momentum in relativistic mechanics[link]

In relativistic mechanics, in order to be conserved, the momentum of an object must be defined as

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{p} = \gamma m_0\mathbf{v}\,,

where m₀ is the invariant mass of the object and γ is the Lorentz factor, given by

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

where v is the speed of the object and c is the speed of light. The inverse relation is given by:^[16]

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{v} = \frac{c^2 \mathbf{p}}{\sqrt{(pc)^2 + (m_0 c^2)^2}} = \frac{c^2 \mathbf{p}}{E} \,,

where Failed to parse (Missing texvc executable; please see math/README to configure.): p = \sqrt{p_x^2 + p_y^2 + p_z^2}

is the magnitude of the momentum.

Relativistic momentum can also be written as invariant mass times the object's proper velocity, defined as the rate of change of object position in the observer frame with respect to time elapsed on object clocks (i.e. object proper time). Within the domain of classical mechanics, relativistic momentum closely approximates Newtonian momentum: at low velocity, γm₀v is approximately equal to m₀v, the Newtonian expression for momentum.

A graphical representation of the interrelation of relativistic energy E, invariant mass m₀, relativistic momentum p, and relativistic mass m = γm₀.

The total energy E of a body is related to the relativistic momentum p by

Failed to parse (Missing texvc executable; please see math/README to configure.): E^2 = (pc)^2 + (m_0c^2)^2\,,

where p denotes the magnitude of p. This relativistic energy-momentum relationship holds even for massless particles such as photons; by setting m₀ = 0 it follows that

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

For both massive and massless objects, relativistic momentum is related to the de Broglie wavelength λ by

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

where h is the Planck constant.

Four-vector formulation[link]

Relativistic four-momentum as proposed by Albert Einstein arises from the invariance of four-vectors under Lorentzian translation. The four-momentum P is defined as:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{P} := (E/c, p_x , p_y ,p_z)\,,

where E = γm₀c² is the total relativistic energy of the system, and p_x, p_y, and p_z represent the x-, y-, and z-components of the relativistic momentum, respectively.

The magnitude ||P|| of the momentum four-vector is equal to m₀c, since

Failed to parse (Missing texvc executable; please see math/README to configure.): ||\mathbf{P}||^2 = (E/c)^2 - p^2 = (m_0c)^2\,.

which is invariant across all reference frames. For a closed system, the total four-momentum is conserved, which effectively combines the conservation of both momentum and energy into a single equation. For example, in the radiationless collision of two particles with rest masses Failed to parse (Missing texvc executable; please see math/README to configure.): m_1

and Failed to parse (Missing texvc executable; please see math/README to configure.): m_2
with initial velocities Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{v}_1
and Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{v}_2

, the respective final velocities Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{v}_3

and Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{v}_4
may be found from the conservation of four-momentum which states that:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{P}_1+\mathbf{P}_2=\mathbf{P}_3+\mathbf{P}_4\,,

where

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{P}_i=m_i\gamma_i(c,\mathbf{v}_i).

For elastic collisions, the rest masses remain the same (Failed to parse (Missing texvc executable; please see math/README to configure.): m_1=m_3

and Failed to parse (Missing texvc executable; please see math/README to configure.): m_2=m_4

), while for inelastic collisions, the rest masses will increase after collision due to an increase in their heat energy content. The conservation of four-momentum can be shown to be the result of the homogeneity of space–time.

Generalization of momentum[link]

Momentum is the Noether charge of translational invariance. As such, not just particles, but fields and other things can have momentum. However, where space–time is curved there is no Noether charge for translational invariance.

Momentum in quantum mechanics[link]

In quantum mechanics, momentum is defined as an operator on the wave function. The Heisenberg uncertainty principle defines limits on how accurately the momentum and position of a single observable system can be known at once. In quantum mechanics, position and momentum are conjugate variables.

For a single particle described in the position basis the momentum operator can be written as

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{p}={\hbar\over i}\nabla=-i\hbar\nabla\,,

where ∇ is the gradient operator, ħ is the reduced Planck constant, and i is the imaginary unit. This is a commonly encountered form of the momentum operator, though the momentum operator in other bases can take other forms. For example, in the momentum basis the momentum operator is represented as

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

where the operator p acting on a wave function ψ(p) yields that wave function multiplied by the value p, in an analogous fashion to the way that the position operator acting on a wave function ψ(x) yields that wave function multiplied by the value x.

Momentum in electromagnetism[link]

Electric and magnetic fields possess momentum regardless of whether they are static or they change in time.^[17] The pressure, P, of an electrostatic (magnetostatic) field upon a metal sphere, cylindrical capacitor or ferromagnetic bar is:

Failed to parse (Missing texvc executable; please see math/README to configure.): P_{static} = {W}= \left[ {\epsilon_0 \epsilon}{\frac{{\mathbf E}^2 }{ {2}}} +{\frac{ 1 }{ {\mu_0 \mu} }} {\frac{{\mathbf B}^2}{{2}}} \right]\,,

where Failed to parse (Missing texvc executable; please see math/README to configure.): {W}\, , Failed to parse (Missing texvc executable; please see math/README to configure.): {\mathbf E}\, , Failed to parse (Missing texvc executable; please see math/README to configure.): {\mathbf B}\, , are the electromagnetic energy density, electric field, and magnetic field respectively. The electromagnetic pressure Failed to parse (Missing texvc executable; please see math/README to configure.): {P}={W}\,

may be sufficiently high to explode the capacitor.

Thus^{[clarification needed]} electric and magnetic fields do carry momentum.

Light (visible, UV, radio) is an electromagnetic wave and also has momentum. Even though photons (the particle aspect of light) have no mass, they still carry momentum. This leads to applications such as the solar sail. The calculation of the momentum of light within dielectric media is somewhat controversial (see Abraham–Minkowski controversy [2]).

Momentum is conserved in an electrodynamic system (it may change from momentum in the fields to mechanical momentum of moving parts). The treatment of the momentum of a field is usually accomplished by considering the so-called energy-momentum tensor and the change in time^{[citation needed]} of the Poynting vector integrated over some volume. This is a tensor field which has components related to the energy density and the momentum density.

The definition of canonical momentum corresponding to the momentum operator of quantum mechanics when it^{[clarification needed]} interacts with the electromagnetic field is, using the principle of least coupling^{[citation needed]}:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf P = m\mathbf v + q\mathbf A\,

, instead of the customary

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf p = m\mathbf v\,,

where:

A is the electromagnetic vector potential

m the charged particle's invariant mass

v its velocity

q its charge.

Non standard terminology is sometimes used for these momenta:^[18] P for canonical momentum, Π = mv for kinetic momentum, and q A for potential momentum.

Analogies between heat, mass, and momentum transfer[link]

There are some notable similarities in equations for momentum, heat, and mass transfer.^[19] The molecular transfer equations of Newton's law for fluid momentum, Fourier's law for heat, and Fick's law for mass are very similar. A great deal of effort has been devoted to developing analogies among these three transport processes so as to allow prediction of one from any of the others.

See also[link]

• Angular momentum
• Conservation law
• Force
• Impulse
• Kinetic energy
• Moment map
• Noether's theorem
• Planck momentum
• Velocity
• Galilean cannon
• Variable-mass system

Notes[link]

References[link]

• Halliday, David; Robert Resnick (1960-2007). Fundamentals of Physics. John Wiley & Sons. Chapter 9. 
• Serway, Raymond; Jewett, John (2003). Physics for Scientists and Engineers (6 ed.). Brooks Cole. ISBN 0-534-40842-7
• Stenger, Victor J. (2000). Timeless Reality: Symmetry, Simplicity, and Multiple Universes. Prometheus Books. Chpt. 12 in particular.
• Tipler, Paul (1998). Physics for Scientists and Engineers: Vol. 1: Mechanics, Oscillations and Waves, Thermodynamics (4th ed.). W. H. Freeman. ISBN 1-57259-492-6
• Hand, Louis N.; Finch, Janet D.. Analytical Mechanics. Cambridge University Press. Chapter 4. 

External links[link]

Look up momentum in Wiktionary, the free dictionary.

• Conservation of momentum – A chapter from an online textbook

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

This article is in a list format that may be better presented using prose. You can help by converting this article to prose, if appropriate. Editing help is available. (May 2010)

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]