= Impulse-momentum theorem
{title2=$\mathbf J_{\mathrm{ext}}=\Delta\mathbf P$}
Integrating the closed-collection balance in <momentum conservation> and applying the <fundamental theorem of calculus> gives
$$
\mathbf J_{\mathrm{ext}}
=\int_{t_i}^{t_f}\sum_i\mathbf F_i^{\mathrm{ext}}(t)\,dt
=\mathbf P(t_f)-\mathbf P(t_i).
$$
For one fixed-<mass> <point mass>, this becomes $\mathbf J_{\mathrm{net}}=\Delta\mathbf p=m\Delta\mathbf v$. Zero external <impulse> implies equal initial and final total <momenta>; <momentum conservation> throughout the interval requires zero net external <force> throughout it.
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