Question

A balloon and its content having mass M is moving up with an acceleration a. The mass that must be released from the content so that the balloon starts moving up with an acceleration '3a' will be

Answer 1

$\frac{2aM}{g+3a}$

Answer 2

Initial state: $F_B - Mg = Ma \implies F_B = M(g+a)$

Final state: $F_B - (M-m)g = (M-m)3a$

Substituting $F_B$: $M(g+a) - (M-m)g = 3a(M-m)$

$Mg + Ma - Mg + mg = 3aM - 3am$

$Ma + mg = 3aM - 3am$

$m(g+3a) = 2aM$

$m = \frac{2aM}{g+3a}$