Answer
a) The total horizontal momentum of the boat plus the person does not change.
b) The speed of the boat decreases.
Work Step by Step
a) The person jumps straight down, meaning he only adds his vertical momentum to the vertical momentum of the boat. Therefore, the total horizontal momentum of the boat plus the person does not change.
b) Take the mass of the boat to be $M$ and the mass of the person to be $m$. Before the jump, the boat has horizontal velocity $V_0$ while the person has horizontal velocity $v=0$
After the jump, the boat and the person become a system, whose mass is $M+m$ and whose horizontal velocity is $V_f$
As shown in a), $\sum p_{x}$ is conserved, so $$MV_0+0=(M+m)V_f$$ $$\frac{V_f}{V_0}=\frac{M}{M+m}\lt1$$ $$V_f\lt V_0$$
The speed of the boat, as we can see, decreases.
