How much force does a 12 gauge exert on your shoulder when firing a deer slug?

Mass of gun = 3.5kg

Mass of slug = 0.22kg

$\vec{F}_{net}$ on shoulder

${\Delta t} \longrightarrow 1/24s$ - 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.

System: Gun + Slug

Surroundings: Nothing

$\vec{F}_{net} = \dfrac{\Delta\vec{p}}{\Delta t}$

$\vec{p}_{sys,f} = \vec{p}_{sys,i}$

$\vec{p}_{1,f} + \vec{p}_{2,f} = \vec{p}_{1,i} + \vec{p}_{2,i}$

$m_1\vec{v}_{1,f} + m_2\vec{v}_{2,f} = m_1\vec{v}_{1,i} + m_2\vec{v}_{2,i}$

${\Delta p_x} = 0$

$P_{tot,x} = 0$

Because

$P_{tot,i,x} = 0$

$0 = M_G * V_G + M_S * V_S \longrightarrow M_G * V_G$ is negative and $M_S * V_S$ is positive$