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

Step 1: Force Equation for Initial Condition 

Let the force \( F \) be the force acting on the balloon. The force equation for the initial condition (with mass \( m \)) is: \[ F - mg = ma \]

Step 2: Force When Mass \( x \) is Released

The force when the mass \( x \) is released becomes: \[ F = ma + mg \]

Step 3: Force After Releasing Mass \( x \)

After releasing mass \( x \), the equation becomes: \[ F - (m - x)g = (m - x) 3a \]

Step 4: Substitute the Value of \( F \)

Substituting the value of \( F \) from the previous equation: \[ Ma + mg - mg + xg = 3ma - 3xa \]

Step 5: Solve for \( x \)

Solving for \( x \): \[ x = \frac{2ma}{g + 3a} \]

Final Answer: \[ x = \frac{2ma}{g + 3a} \]