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 the force \( F \) be the force acting on the balloon. The force equation for the initial condition (with mass \( m \)) is: \[ F - mg = ma \] The force when the mass \( x \) is released becomes: \[ F = ma + mg \] After releasing mass \( x \), the equation becomes: \[ F - (m - x)g = (m - x) 3a \] Substituting the value of \( F \) from the previous equation: \[ Ma + mg - mg + xg = 3ma - 3xa \] Solving for \( x \): \[ x = \frac{2ma}{g + 3a} \]