Question

a balloon of mass m is rising up with an acceleration a . show that the fraction of mass of balloon that must be detached in order to double its acceleration is ma/g + 2a.

Answer 1

$$\begin{gathered} \mathrm{Let} \\ \mathrm{R}=\mathrm{Upthrust}\mathrm{on}\mathrm{the}\mathrm{balloon} \\ \mathrm{Case}\mathrm{I}: \\ \mathrm{ma}=\mathrm{R}-\mathrm{mg}......\left[1\right] \\ \mathrm{Case}\mathrm{II}: \\ \mathrm{Let} \\ \mathrm{m}\text{'}=\mathrm{Mass}\mathrm{detached} \\ \left[\mathrm{m}-\mathrm{m}\text{'}\right]2\mathrm{a}=\mathrm{R}-\left[\mathrm{m}-\mathrm{m}\text{'}\right]\mathrm{g}......\left[2\right] \\ \mathrm{From}\left(1\right)\mathrm{and}\left(2\right) \\ \left[\mathrm{m}-\mathrm{m}\text{'}\right]2\mathrm{a}=\mathrm{ma}+\mathrm{mg}-\left[\mathrm{m}-\mathrm{m}\text{'}\right]\mathrm{g} \\ \left[\mathrm{m}-\mathrm{m}\text{'}\right]\left[\mathrm{g}+2\mathrm{a}\right]=\mathrm{m}\left[\mathrm{a}+\mathrm{g}\right] \\ \mathrm{m}-\mathrm{m}\text{'}=\frac{\mathrm{m}\left[\mathrm{g}+\mathrm{a}\right]}{\mathrm{g}+2\mathrm{a}} \\ \mathrm{m}\text{'}=\mathrm{m}\left[1-\frac{\mathrm{g}+\mathrm{a}}{\mathrm{g}+2\mathrm{a}}\right] \\ =\frac{\mathbf{ma}}{\mathbf{g}\mathbf{+}\mathbf{2}\mathbf{a}} \end{gathered}$$