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:
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- \( \frac{2Ma}{3a + g} \)
- \( \frac{3Ma}{2a - g} \)
- \( \frac{3Ma}{2a + g} \)
- \( \frac{2Ma}{3a - g} \)
The Correct Option is A
Solution and Explanation
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: \[ x = \frac{2ma}{g + 3a} \]
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