Question

A hot air balloon is carrying some passengers, and a few sandbags of mass $1\, kg$ each so that its total mass is $480 \,kg$. Its effective volume giving the balloon its buoyancy is $V$. The balloon is floating at an equilibrium height of $100\, m$. When $N$ number of sandbags are thrown out, the balloon rises to a new equilibrium height close to $150\, m$ with its volume $V$ remaining unchanged. If the variation of the density of air with height $h$ from the ground is $\rho( h )=\rho_{0} e ^{-\frac{ h }{ h_{0}}}$, where $\rho_{0}=1.25 \,kg \,m ^{-3}$ and $h _{0}=6000\, m$, the value of $N$ is _______.

Answer 1

$\frac{480-N}{480}=\frac{\rho_{2}}{\rho_{1}}$
$1-\frac{N}{480}=e^{-\left(\frac{150-100}{6000}\right)}$
$\Rightarrow 1-\frac{N}{480}=1-\frac{5}{600}$
$N=4$