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Example: Firing a deer slug.
How much force does a 12 gauge exert on your shoulder when firing a deer slug?
Facts
Mass of gun = 3.5kg
Mass of slug = 0.22kg
Lacking
\vec{F}_{net} on shoulder
Approximations & Assumptions
${\Delta t} \longrightarrow 1/24s$ - Based on when a gun is fired in a movie, it usually occurs at about one movie frame, therefore, the collision time is less than 1/24s.
$\vec{V}_{Slug} \longrightarrow 500m/s$ This is a conservative estimate based on an internet search.
Representations
System: Gun + Slug
Surroundings: Nothing
$\vec{F}_{net} = \dfrac{\Delta\vec{p}}{\Delta t}$
$\vec{p}_{sys,f} = \vec{p}_{sys,i}$
$\vec{p}_{1,f} + \vec{p}_{2,f} = \vec{p}_{1,i} + \vec{p}_{2,i}$
$m_1\vec{v}_{1,f} + m_2\vec{v}_{2,f} = m_1\vec{v}_{1,i} + m_2\vec{v}_{2,i}$