# Definition:Linear Momentum

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## Definition

The **linear momentum** of a body is its mass multiplied by its velocity.

- $\mathbf p = m \mathbf v$

As mass is a scalar quantity and velocity is a vector quantity, it follows that linear momentum is a vector quantity.

### Dimension

The dimension of measurement of **linear momentum** is:
$M L T^{-1}$.

### Relativistic Model

A more accurate model for the linear momentum of a body is given by:

- $\mathbf p = \gamma m \mathbf v$

where $\gamma$ is the Lorentz Factor:

- $\gamma = \dfrac c {\sqrt {c^2 - v^2} } = \dfrac 1 {\sqrt {1 - v^2 / c^2} }$

where:

- $c$ is the speed of light in vacuum
- $v$ is the magnitude of $\mathbf v$: $v = \size {\mathbf v}$

It is clear $\gamma \approx 1$ (and thus that $\mathbf p \approx m \mathbf v$) for values of $v$ much less than $c$.

## Also known as

**Linear momentum** is frequently referred to as just **momentum**.

## Also see

## Sources

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{VI}$: On the Seashore - 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 22$: Vectors and Scalars - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.7$: A Simple Approach to $E = M c^2$ - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**momentum (linear momentum)**(*plural***momenta**)