Answer
a) The character recoils at a velocity of $-0.14m/s$
b) The character recoils at a velocity of $-7.06\times10^{-3}m/s$
Work Step by Step
a) At the start, the character is stationary, so $\sum \vec{p_0}=0$
Then, she fires a bullet with mass $m=0.01kg$ and velocity $v=+720m/s$. The character herself, whose mass $M=51kg$, recoils with velocity $V$, which needs to be found.
Since we assume no external force, the total linear momentum is conserved. Therefore,
$$m\vec{v}+M\vec{V}=\sum \vec{p_0}=0$$ $$\vec{V}=-\frac{m\vec{v}}{M}=-0.14m/s$$
b) Similarly, the total linear momentum is still conserved, so $$m\vec{v}+M\vec{V}=\sum \vec{p_0}=0$$ $$\vec{V}=-\frac{m\vec{v}}{M}$$
except now $m=5\times10^{-4}kg$, so $$\vec{V}=-7.06\times10^{-3}m/s$$
