Your friend wants to start a business giving hot-air balloon rides. The empty balloon, the basket and the occupants have a total maximum mass of 1000 kg. If the balloon itself has a diameter of 22.0 m when fully inflated with hot air. Estimate the required density of the hot air. (Take the density of air as 1.293 kg/m $^{3}$ . Neglect the buoyancy force on the basket and people).

2.98 $k g / m^{3}$

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5.44 $k g / m^{3}$

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2.38 $k g / m^{3}$

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1.11 $k g / m^{3}$

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Solution

The correct option is D
1.11 $k g / m^{3}$

We can use Archimedes' principle and the condition for vertical equilibrium to estimate the density of the hot air that would enable the balloon and its payload to lift off.
$F_{b u o y a n t} = B = F_{g r a v i t a t i o n a l} = m g (1)$
Using Archimedes' principle, express the buoyant force B:
$B = W_{d i s p l a c e d f l u i d} = m_{d i s p l a c e d f l u i d} g$
$= ρ_{a i r} V_{b a l l o o n} g$
Substituting for B in equation (1)
$ρ_{a i r} V_{b a l l o o n} g = m g \Rightarrow ρ_{a i r} V_{b a l l o o n} = m (2)$
The total mass m to be lifted is the sum of the mass of the payload $m_{p}$ and the mass of the hot air:
$m = m_{p} + m_{h o t a i r} = m_{p} + ρ_{h o t a i r} V_{b a l l o o n}$
Substitute for m in equation (2) to obtain:
$ρ_{a i r} V_{b a l l o o n} = m_{p} + ρ_{h o t a i r} V_{b a l l o o n}$
Solving for $ρ_{h o t a i r}$ and simplifying yields:
$ρ_{h o t a i r} = \frac{ρ_{a i r} V_{b a l l o o n} - m_{p}}{V_{b a l l o o n}}$

Substitute numerical values and evaluate $ρ_{h o t a i r}$
$ρ_{h o t a i r} = 1.11 k g / m^{3}$

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