Linear momentum

Definition

For a single body in pure translation

The linear momentum of a body is the vector $m{\overline {v}}$, where $m$ is the mass of the body and ${\overline {v}}$ is the velocity of the body.

For a system of bodies, each in pure translation

The linear momentum of a system is the sum of the linear momenta of all the bodies in that system. If the system comprises $n$ bodies with masses $m_{1},m_{2},\dots ,m_{n}$ respectively and velocities ${\overline {v}}_{1},{\overline {v}}_{2},\dots ,{\overline {v}}_{n}$ respectively, the total linear momentum is given by:

$\!\sum _{i=1}^{n}m_{i}{\overline {v}}_{i}$

This is a vector summation.

For a single body undergoing motion that is not purely translation

In this case, we integrate the velocity vector over a mass differential:

$\int {\overline {v}}\,dm$

This is equivalent to integrating the product of velocity and density over a volume differential:

$\int {\overline {v}}\rho \,dV$

Units and dimensions

The dimensions of linear momentum are $MLT^{-1}$, and the SI units are $kgm/s$ (kilogram meters per second). These are the same as the units of impulse, which is related to linear momentum via a formulation of Newton's second law: the net external impulse on a system equals the change in its linear momentum.