{title2=$\mathbf J$}
{wiki}

The <impulse> of a <force> over a <time> interval is its <vector> <integral>:
$$
\mathbf J=\int_{t_i}^{t_f}\mathbf F(t)\,dt.
$$
Its <International System of Units> unit is $\mathrm{N\,s}=\mathrm{kg\,m/s}$. The <impulse-momentum theorem> equates the net <impulse> to the change in <momentum>. For one fixed-<mass> <point mass>, integrate its net <force>; for a closed collection obeying <Newton's third law>, integrate the sum of external <forces>. In one dimension, the signed area under the <force> versus <time> graph gives the <impulse>; negative <force> contributes negative area. https://openstax.org/books/university-physics-volume-1/pages/9-2-impulse-and-collisions[OpenStax University Physics Volume 1, section 9.2: Impulse and Collisions]{ref}

Three stipulated rightward net <force> pulses illustrate the same $J=6\,\mathrm{N\,s}$: constant $12\,\mathrm N$ for $0.5\,\mathrm s$, a triangle rising from zero to $24\,\mathrm N$ at $0.25\,\mathrm s$ and falling to zero at $0.5\,\mathrm s$, and constant $3\,\mathrm N$ for $2\,\mathrm s$. Their areas are
$$
(12\,\mathrm N)(0.5\,\mathrm s)
=\frac12(24\,\mathrm N)(0.5\,\mathrm s)
=(3\,\mathrm N)(2\,\mathrm s)
=6\,\mathrm{N\,s}.
$$
Applied separately to the same fixed <mass> $m=2\,\mathrm{kg}$, each gives $\Delta v=J/m=3\,\mathrm{m/s}$. Their peak <forces> differ: $12$, $24$, and $3\,\mathrm N$. Thus <impulse> fixes the final <momentum> change, while the <force> history determines how abruptly it occurs.

\Image[/impulse-force-pulses.png]
{title=Three force pulses have the same impulse but different durations and peak forces}

The mean net <force> over the chosen interval is $\overline{\mathbf F}=\mathbf J/(t_f-t_i)$. Extending the interval for a fixed <impulse> reduces this mean <force>. A lower peak <force> additionally depends on the pulse shape: the two short pulses above have the same mean $12\,\mathrm N$ but different peaks. This explains why controlling the duration and shape of a stopping <impulse> matters in cushioning a <collision>.


 Impulse (source code)