Linear momentum

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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

Question Answer
Scalar or vector? Vector
Instantaneous or time-cumulative Instantaneous
MLT dimensions MLT^{-1}: MLT;1;1;-1 (same as those of impulse)
SI units kgm/s (kilograms meter per second) or N-s (Newton-second)