A balloon with mass $m$ is descending down with an acceleration $a$ (where $a

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 $a$?

Accepted Answer

1. **Case 1 (Descending):** 
   - Let the upthrust (buoyant force) be $B$.
   - Forces acting on the balloon: Weight $mg$ (downwards) and Upthrust $B$ (upwards).
   - Since it is accelerating downwards with $a$, the net force equation is:
     $$mg - B = ma \implies B = m(g - a)$$ ... (i)
     (Reference: NCERT Class 11, Physics Part I, Chapter 5, Newton's Second Law).

2. **Case 2 (Ascending):**
   - Let mass $\Delta m$ be removed. The new mass is $(m - \Delta m)$.
   - The upthrust $B$ remains the same (assuming volume doesn't change significantly).
   - The balloon accelerates upwards with $a$. The net force equation is:
     $$B - (m - \Delta m)g = (m - \Delta m)a$$

3. **Calculation:**
   - Substitute $B$ from equation (i) into the second equation:
     $$m(g - a) - (m - \Delta m)g = (m - \Delta m)a$$
     $$mg - ma - mg + \Delta mg = ma - \Delta ma$$
     $$-ma + \Delta mg = ma - \Delta ma$$
     $$\Delta m(g + a) = 2ma$$
     $$\Delta m = \frac{2ma}{g + a}$$

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