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?

When the balloon is descending down with acceleration a,mg - B = m × a       .....(i)where, B = Buoyant force.Here, we should assume that while removing some mass, the volume of balloon and hence, buoyant force will not change.Let the new mass of the balloon is m'.So, mass removed = m - m'When the balloon is moving up with acceleration a, thenB - m'g = m' × a       .....(ii)Adding Eqs. (i) and (ii), we get⇒  mg-m'g=ma+m'a⇒  (mg-ma)=m'(g+a)⇒  m(g-a)=m'(g+a)∴  m'=m(g-a)(g+a)So, mass removed,Δm=m-m'=m1-(g-a)(g+a)=m(g+a)-(g-a)(g+a)=mg+a-g+ag+a=2mag+a