A balloon with mass $m$ is descending down with an acceleration $a$ (where $a

Question

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 $2a$?"

Professor PC Thomas Classes · Answer

Let $F$ be upthrust. Descending: $mg - F = ma \implies F = m(g-a)$. Let mass $\Delta m$ be removed. Ascending with $2a$: $F - (m - \Delta m)g = (m - \Delta m)(2a)$. Substituting $F$: $m(g-a) - (m-\Delta m)g = (m - \Delta m)2a \implies mg - ma - mg + \Delta m g = 2ma - 2\Delta m a \implies \Delta m(g + 2a) = 3ma \implies \Delta m = \frac{3ma}{g+2a}$.