A balloon of mass $M$ descends with an acceleration $a_{0}$ The mass that must be thrown out in order to give the balloon an equal upward acceleration will be

\frac{M a_{0}}{g}

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\frac{2 M a_{0}}{g}

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\frac{2 M a_{0}}{g + a_{0}}

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\frac{M (g + a_{0})}{a_{0}}

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Solution

The correct option is B $\frac{2 M a_{0}}{g + a_{0}}$
When the balloon descends with acceleration $a_{o}$ , total downward force is equal to $M a_{o}$ .
$M a_{o} =$ net downward force $=$ weight of balloon - buoyant force
$M a_{o} = M g - V_{b} ρ_{a} g . . . . . (I)$
Now lets say $m$ mass is removed from the balloon, due to which the balloon rises with $a_{o}$ , So the net upward force this time is equal to $(M - m) a_{o}$
$(M - m) a_{o} =$ net upward force $=$ buoyant force- net weight of balloon
$(M - m) a_{o} = V_{b} ρ_{a} g - (M - m) g . . . . . (I I)$
adding $(I)$ and $(I)$ , we have
$2 M a_{o} - m a_{o} = m g \Rightarrow 2 M a_{o} = m g + m a_{o} = m (g + a_{o}) \Rightarrow m = \frac{2 M a_{o}}{(g + a_{o})}$

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A balloon of mass M descends with an acceleration a0 The mass that must be thrown out in order to give the balloon an equal upward acceleration will be

A balloon of mass $M$ descends with an acceleration $a_{0}$ The mass that must be thrown out in order to give the balloon an equal upward acceleration will be