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(A) Let the upthrust of air be $F_a$.For the downward motion of the balloon with mass $m$ and acceleration $a$:$mg - F_a = ma$  --- $(1)$From this,we

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$\frac{2ma}{g + a}$

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(A) Let the upthrust of air be $F_a$.For the downward motion of the balloon with mass $m$ and acceleration $a$:$mg - F_a = ma$  --- $(1)$From this,we get $F_a = m(g - a)$.Now,let the mass removed be $\Delta m$. The new mass of the balloon is $(m - \Delta m)$.For the upward motion with acceleration $a$:$F_a - (m - \Delta m)g = (m - \Delta m)a$$F_a = (m - \Delta m)(g + a)$ --- $(2)$Equating the expressions for $F_a$ from $(1)$ and $(2)$:$m(g - a) = (m - \Delta m)(g + a)$$mg - ma = mg + ma - \Delta m(g + a)$$\Delta m(g + a) = 2ma$$\Delta m = \frac{2ma}{g + a}$

$A$ balloon with mass $m$ is descending down with an acceleration $a$ (where $a < g$). How much mass should be removed from it so that it starts moving up with an acceleration $a$?

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