# Momentum

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For the Sci-Fi Channel movie, see Momentum (movie)

In physics, momentum is the product of the velocity and mass of an object.

## Introduction

If an object is moving in any reference frame, then it has momentum in that frame. The amount of momentum that an object has depends on two variables: the mass and the velocity of the moving object in the frame of reference. This can be written as:

momentum = mass × velocity

In physics, the symbol for momentum is a small p; so the above equation can be rewritten as:

p = m v

where m is the mass and v the velocity. The SI unit of momentum is kilogram metres per second (kg m/s). The velocity of an object is given by its speed and its direction. Because momentum depends on velocity, it too has a magnitude and a direction: it is a vector quantity. For example the momentum of a 5 kg bowling ball would have to be described by the statement that it was moving westward at 2 m/s. It is insufficient to say that the ball has 10 kg m/s of momentum; the momentum of the ball is not fully described until information about its direction is given.

A step change in an object's momentum is known as an impulse:

The impulse (mass × change in velocity) = force applied × the time over which the force was applied.

## Origin of momentum

Momentum arises from the condition that an experiment must give the same results regardless of the position or relative velocity of the observer. More formally, it is the requirement of invariance under translation. Classical momentum is the result of this invariance in three dimensions. The definition of momentum was changed when Einstein formulated special relativity, so that its magnitude would remain invariant under relativistic transformations.

## Conservation of momentum

Because of the way it is defined, momentum is always conserved. In the absence of external forces, a system will have constant momentum: a property that is identical to Newton's law of inertia, his first law of motion. Newton's third law of motion, the law of reciprocal actions, dictates that the forces acting between systems are equal, which is equivalent to a statement of the conservation of momentum.

### Conservation of momentum and collisions

Momentum has the special property that it is always conserved, even in collisions. 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 unknown velocities following a collision.

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 momentum before the collision must equal the sum of the momentum after the collision:

${\displaystyle m_{1}\mathbf {v} _{1,i}+m_{2}\mathbf {v} _{2,i}=m_{1}\mathbf {v} _{1,f}+m_{2}\mathbf {v} _{2,f}\,}$
where the subscript i signifies initial, before the collision, and f signifies final, after the collision.

Usually, we either only know the velocities before or after a collision and like to also find out the opposite. Correctly solving this problem means you have to know what kind of collision took place. There are two basic kinds of collisions, both of which conserve momentum:

#### Elastic collisions

A collision between two pool or snooker balls is a good example of an almost totally elastic collision. 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:

${\displaystyle {\begin{matrix}{\frac {1}{2}}\end{matrix}}m_{1}v_{1,i}^{2}+{\begin{matrix}{\frac {1}{2}}\end{matrix}}m_{2}v_{2,i}^{2}={\begin{matrix}{\frac {1}{2}}\end{matrix}}m_{1}v_{1,f}^{2}+{\begin{matrix}{\frac {1}{2}}\end{matrix}}m_{2}v_{2,f}^{2}\,}$

Since the 1/2 factor is common to all the terms, it can be taken out right away.

In the case of two objects colliding head on we find that the final velocity

${\displaystyle v_{1,f}=\left({\frac {m_{1}-m_{2}}{m_{1}+m_{2}}}\right)v_{1,i}+\left({\frac {2m_{2}}{m_{1}+m_{2}}}\right)v_{2,i}\,}$

${\displaystyle v_{2,f}=\left({\frac {2m_{1}}{m_{1}+m_{2}}}\right)v_{1,i}+\left({\frac {m_{2}-m_{1}}{m_{1}+m_{2}}}\right)v_{2,i}\,}$

#### Inelastic collisions

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

${\displaystyle m_{1}\mathbf {v} _{1,i}+m_{2}\mathbf {v} _{2,i}=\left(m_{1}+m_{2}\right)\mathbf {v} _{f}\,}$

## Changes in momentum

Although momentum is conserved within a closed system, individual parts of a system can undergo changes in momentum. In classical mechanics, an impulse changes the momentum of a body, and has the same units and dimensions as momentum. The SI unit of impulse is the same as for momentum (kg m/s). An impulse is calculated as the integral of force with respect to time.

${\displaystyle \mathbf {I} =\int \mathbf {F} \,dt}$

where

I is the impulse, measured in kilogram metres per second
F is the force, measured in newtons
t is the time duration, measured in seconds

In the presence of a constant force, impulse is often written using the formula

${\displaystyle \mathbf {I} =\mathbf {F} \Delta t}$

where

${\displaystyle \Delta t}$ is the time interval over which the force (F) is applied.

Using the definition of force yields:

${\displaystyle \mathbf {I} =\int {\frac {d\mathbf {p} }{dt}}\,dt}$
${\displaystyle \mathbf {I} =\int d\mathbf {p} }$
${\displaystyle \mathbf {I} =\Delta \mathbf {p} }$

It is therefore common to define impulse as a change in momentum.

## Momentum in relativistic mechanics

Relativistic momentum as proposed by Albert Einstein arises from the invariance of four-vectors under Lorentzian translation. These four-vectors appear spontaneously in the Green's function from quantum field theory.

A vector, called the Four-momentum is defined as:

[E/c p]

where E is the total energy of the system, and p is called the "relativistic momentum" defined thus:

${\displaystyle E=\gamma mc^{2}\;}$
${\displaystyle \mathbf {p} =\gamma m\mathbf {v} }$

where

${\displaystyle \gamma ={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}}$
where
${\displaystyle v^{2}=\mathbf {v} \cdot \mathbf {v} }$.

Setting velocity to zero, one derives that the rest mass and the energy of an object are related by E=mc².

The "length" of the vector that remains constant is defined thus:

${\displaystyle \mathbf {p} \cdot \mathbf {p} -E^{2}/c^{2}}$

Massless objects such as photons also carry momentum; the formula is p=E/c, where E is the energy the photon carries and c is the speed of light.

Momentum is the Noether charge of translational invariance. As such, even fields as well as other things can have momentum, not just particles. However, in curved space-time which is not asymptotically Minkowski, momentum isn't defined at all.

## Momentum in quantum mechanics

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 interchangeable.

For a single particle with no electric charge and no spin, the momentum operator can be written in the position basis as

${\displaystyle \mathbf {p} ={\hbar \over i}\nabla =-i\hbar \nabla }$

where ${\displaystyle \nabla }$ is the gradient operator. This is a commonly encountered form of the momentum operator, though not the most general one.

## Figurative use

A process may be said to gain momentum. The terminology implies that it requires effort to start such a process, but that it is relatively easy to keep it going. Alternatively, the expression can be seen to reflect that the process is adding adherents, or general acceptance, and thus has more mass at the same velocity; hence, it gained momentum.