Question

A balloon is descending with a constant acceleration $a$ which is less than the acceleration of gravity $g$. The weight of the ballon, with its basket and contents is $W$. What weight $w$, should be dropped from the balloon so that it will ascend up with acceleration $a$? Neglect air resistance and assume that a constant force $F$ is acting on balloon in upward direction.

Answer 1

Step 1: Given data

1. A balloon descending with a constant acceleration $a$, for the weight of balloon with its basket and contents being $W$.
2. A balloon ascending with a constant acceleration $a$, for the weight of the balloon being $w$.

Step 2: Determine the weight $\mathit{w}$ that is thrown from the balloon

1. A balloon descending with a constant acceleration $a$, for the weight of balloon with its basket and contents being $W$.
2. $\Rightarrow Wg-F=Wa$
3. $\Rightarrow F=Wg-Wa$ ……$\left(1\right)$
4. A balloon ascending with a constant acceleration $a$, for the weight of the balloon being $w$.
5. $\Rightarrow F-wg=wa$ ……$\left(2\right)$
6. Substitute equation $\left(1\right)$ in $\left(2\right)$
7. $\Rightarrow \left(Wg-Wa\right)-wg=wa$
8. $\Rightarrow W(g-a)=w(g+a)$
9. On further solving the above equation we get,
10. $\Rightarrow w=W\frac{(g-a)}{(g+a)}$

Therefore, the weight of the balloon that should be dropped is $\mathit{W}\frac{(g-a)}{(g+a)}\mathbf{}\mathit{u}\mathit{n}\mathit{i}\mathit{t}\mathit{s}$.