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 (Take ‘g’ as acceleration due to gravity)

Laws of Motion PYQ — JEE Main 2025 · 28 Jan · Shift 2 · Q57 with Solution

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Correct answer: B

Let the upward buoyant/lifting force be constant and equal to $F$ .

Initially, the balloon of mass $M$ moves upward with acceleration $a$ .

Taking upward as positive, by Newton’s second law: $F - Mg = Ma$ $F = M(g+a)$
After releasing some mass

Let the released mass be $m$ .

Then the new mass of the balloon system becomes: $M-m$

The balloon now moves upward with acceleration $3a$ .

Again applying Newton’s second law: $F - (M-m)g = (M-m)3a$
Substitute $F = M(g+a)$

$M(g+a) - (M-m)g = 3a(M-m)$

Expand the left side: $Mg + Ma - Mg + mg = 3aM - 3am$ $Ma + mg = 3aM - 3am$
Collect terms containing $m$

$mg + 3am = 3aM - Ma$ $m(g+3a) = 2Ma$

Therefore, $m = rac{2Ma}{g+3a}$

Rearranging: $m = rac{2Ma}{3a+g}$
Match with the options

This corresponds to: $\boxed{\frac{2Ma}{3a+g}}$

So the correct option is B.