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 $ML{T}^{-1}$, 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.