Answer
The balloon moves down with a speed of $~~0.5~m/s$
Work Step by Step
Let "up" be the positive direction.
Note that the initial momentum of the balloon-climbing man system is zero before the man starts climbing.
Let $v$ be the balloon's velocity while the man is climbing. Then $(v+2.5~m/s)$ is the man's velocity.
We can use conservation of momentum to find the velocity of the balloon while the man is climbing up:
$p_f = p_i$
$(80~kg)(v+2.5~m/s)+(320~kg)(v) = 0$
$(80~kg)(v)+(320~kg)(v) = -(80~kg)(2.5~m/s)$
$v = -\frac{(80~kg)(2.5~m/s)}{400~kg}$
$v = -0.5~m/s$
The balloon moves down with a speed of $~~0.5~m/s$
