# What is the integral of the momentum function?

##### 1 Answer
Jan 18, 2015

The momentum formula is typically given by $p = m v$, where $p$ is momentum, $m$ is mass, and $v$ is velocity. One must first decide whether one wishes to integrate with respect to velocity or with respect to mass. If one integrates the function with respect to velocity (and thus treats momentum as a function of velocity), one receives:

$\int p \left(v\right) \mathrm{dv} = \int m v \mathrm{dv}$.

If we assume that mass is constant, then we can factor it out:

$m \int v \mathrm{dv}$

Then, by using the power rule with respect to integrals:

$m \int v \mathrm{dv} = m \frac{1}{2} {v}^{2} = \frac{1}{2} m {v}^{2}$

Note that this is equivalent to the formula for kinetic energy.