Question 422: Properties of Matter

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
$$\rho \left( h \right) = {\rho _0}{e^{ - {h \over {{h_0}}}}}$$, where $$\rho $$
0
= 1.25 kg m
−3
and
h
0
= 6000 m, the value of N is _________.

Accepted Answer

Weight = Upthrust
$$mg = {F_u} \Rightarrow 480 	imes 10 = ho Vg$$
$$480 	imes 10 = {ho _0}{e^{ - {h \over {{h_0}}}}} \Rightarrow 480 	imes 10 = {ho _0}{e^{ - {{100} \over {6000}}}}Vg$$ .... (i)
$$(480 - N 	imes 1)10 = ho 'Vg$$
$$(480 - N)10 = {ho _0}{e^{ - {{150} \over {6000}}}}Vg$$ .... (ii)
Dividing Eq. (i) by Eq. (ii), we get
$${{480} \over {480 - N}} = {e^{\left( {{{150 - 100} \over {6000}}} ight)}}$$
$${{480} \over {480 - N}} = {e^{{{50} \over {6000}}}} \Rightarrow {{480} \over {480 - N}} = {e^{{1 \over {120}}}}$$
N = 4

Properties of Matter

Integer / Numeric Answer Type