Question

A recreational (open) hot air spherical balloon has a radius of 7.30 m when fully inflated. The total weight of the balloon, basket, ballast and pilot is 1710 N. When the balloon is released from its mooring, it moves upward and reached a terminal velocity of VT = 0.910 m/s. If the air resistance is given by Fair resistance = b VTwhere b = 2570 kg/s, what is the density in the hot air in the balloon? The density of surrounding air is 1.293 kg/m3.

Answer 1

r = radius of the balloon = 7.30 m

V = Volume of the balloon = 4 $\pi$ r³/3 = 4 (3.14) (7.30)³/3 = 1628.7 m³

$\rho$ _air = density of air = 1.293 kg/m³

$\rho$ _hotair = density of hot air = ?

F_g = total weight of the balloon, basket, ballast and pilot in down direction = 1710 N

V_T = terminal velocity = 0.910 m/s

F_t = force due to air resistance in down direction = b V_T = (2570) (0.910) = 2338.7 N

F_b = force of buoyancy in upward direction = $\rho$ _air V g = (1.293) (1628.7) (9.8) = 20638 N

F_hotgas = weight of the hot gas = $\rho$ _hotair Vg

using equilibrium of force

F_b = F_hotgas + F_t + F_g

inserting the values

20638 = $\rho$ _hotair Vg + 2338.7 + 1710

20638 = $\rho$ _hotair (1628.7 x 9.8) + 2338.7 + 1710

$\rho$ _hotair = 1.04 kg/m³