Question

A balloon with mass m is descending down with an acceleration a (where a < g). How much mass should be removed from it so that it starts moving up with an acceleration $a$?

• A. $\frac{2ma}{g+a}$
• B. $\frac{2ma}{g-a}$
• C. $\frac{ma}{g+a}$
• D. $\frac{ma}{g-a}$

Answer 1

The Correct Option is A

To solve this problem, we need to analyze the forces acting on the balloon and understand the changes needed for it to ascend with an acceleration \(a\).

Step 1: Current Scenario (Descending with Acceleration \(a\))

• When the balloon descends with acceleration \(a\), the net force acting downward is \(mg - T\), where \(T\) is the buoyant force or upthrust acting on the balloon.
• According to Newton's second law, the net force is given by: \( T - mg = -ma \) because the balloon is accelerating downward.
• Rearranging the equation, we get: \( T = m(g - a) \).

Step 2: New Scenario (Ascending with Acceleration \(a\))

• To ascend with acceleration \(a\), the net upward force must be positive.
• Let \(m_f\) be the final mass after removing some mass from the balloon.
• The equation becomes: \(T - m_f g = m_f a\).
• Substituting the expression for \(T\) from Step 1: \(m(g - a) - m_f g = m_f a\).
• Simplifying, we get: \( m(g - a) = m_f (g + a) \).

Step 3: Mass to be Removed

• Solving for \(m_f\), we have: \( m_f = \frac{m(g - a)}{g + a} \).
• The mass removed \(\Delta m\) is: \( \Delta m = m - m_f \).
• Substituting for \(m_f\): \( \Delta m = m - \frac{m(g - a)}{g + a} \).
• Simplifying further: \( \Delta m = m \left( 1 - \frac{g - a}{g + a} \right) \)
• \( \Delta m = m \left( \frac{(g + a) - (g - a)}{g + a} \right) \), simplifying gives:
• \( \Delta m = m \left( \frac{2a}{g + a} \right) \).

Therefore, the mass that should be removed for the balloon to start ascending with acceleration \(a\) is \( \frac{2ma}{g+a} \).

Hence, the correct answer is: \( \frac{2ma}{g+a} \).