Impulse equals Change in Momentum

Theorem

Let a force $\mathbf F$ be applied to a particle $P$.

Then the impulse $\mathbf J$ imparted to $P$ is equal to the change of momentum of $P$.

$\ds \mathbf F$

$=$

$\ds \dfrac {\d \mathbf p} {\d t}$

$\ds \leadsto \ \ $

$\ds \int_{t_1}^{t_2} \mathbf F \rd t$

$=$

$\ds \int_{t_1}^{t_2} \dfrac {\d \mathbf p} {\d t}$

integrating $\mathbf F$ with respect to time between limits $t_1$ to $t_2$

$\ds \leadsto \ \ $

$\ds \mathbf J$

$=$

$\ds \bigintlimits {\mathbf p} {t \mathop = t_1} {t \mathop = t_2}$

$\ds $

$=$

$\ds \mathbf p_{t_2} - \mathbf p_{t_1}$

Hence the impulse $\mathbf J$ equals the change of (linear) momentum between $t_1$ and $t_2$, as required.

$\blacksquare$