### Balloon with Weights

The weight at first is $Mg$. The total force at first is $-Ma$.
The force up at first is $B=Mg-Ma$. We want to have a total force
of $(M-m)a$ and a weight of $(M-m)g$ at the end.
$(M-m)a = B-(M-m)g=Mg-Ma-Mg+mg=mg-Ma$
$m = \frac{2Ma}{a+g}.$