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Q. 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 $a$?

AIPMTAIPMT 2014Laws of Motion

Solution:

Let $F $ be the upthrust of the air.
As the balloon is descending down with an acceleration a,
$\therefore mg=F=ma$ ... (i)
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Let mass $m_0$ be removed from the balloon so that it starts moving up with an acceleration $a$.
Then, $F-(m-m_0)g=(m-m_0)a$
$F-mg+m_0g=ma-m_0a$ ... (ii)
Adding eqn. (i) and eqn. (ii), we get
$m_0g=2ma-m_0a$
$m_0g+m_0a=2\,ma$
$m_0(g+a)=2\,ma$
$m_0=\frac{2ma}{a+g}$