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:

• A. \( \frac{3Ma}{2a + g} \)
• B. \( \frac{3Ma}{2a - g} \)
• C. \( \frac{2Ma}{3a + g} \)
• D. \( \frac{2Ma}{3a - g} \)

Hint:

In problems involving forces and accelerations, remember to apply Newton’s second law for both the initial and final conditions, and use the relationship between mass and acceleration carefully.

Answer 1

The Correct Option is A

Let \( F \) denote the force exerted on the balloon. The initial force equation, considering mass \( m \), is: \[ F - mg = ma \] Upon release of mass \( x \), the force equation modifies to: \[ F = ma + mg \] Following the release of mass \( x \), the governing equation is: \[ F - (m - x)g = (m - x) 3a \] Substituting the expression for \( F \) from the preceding equation: \[ Ma + mg - mg + xg = 3ma - 3xa \] Solving for \( x \), we obtain: \[ x = \frac{2ma}{g + 3a} \]