It has been proposed that we could explore Mars using inflated balloons to hover

Question

It has been proposed that we could explore Mars using inflated balloons to hover just above the surface. The buoyancy of the atmosphere would keep the balloon aloft. The density of the Martian atmosphere is 0.0154 kg/m^3 (although this varies with temperature). Suppose we construct these balloons of a thin but tough plastic having a density such that each square meter has a mass of 4.90 g. We inflate them with a very light gas whose mass we can neglect. What should be the radius of these balloons so they just hover above the surface of Mars? What should be the mass of these balloons so they just hover above the surface of Mars? If we released one of the balloons from on earth, where the atmospheric density is 1.20 kg/m^3, what would be its initial acceleration assuming it was the same size as on Mars?

Explanation / Answer

The buoyant force B equals the weight of the air displaced by the balloon .

B = rho_air * V*g

V = 4pi*r^3/3

surface area of sphere 4pi*R^2

mass pf balloon = 4.9 x 10^-3 /m^2 * 4pi*R^2

B = m*gm * rho_air * V*gm = m*gm

rho_air * 4 * pi * R^3/3 = (4.9 x 10^-3 kg/m^2) * (4pi*R^2)

R = 3 * 4.9 x 10^-3 / rho_air

R = 0.9545 m

part b )

m = (4.9 x 10^-3 kg/m^2) * ( 4pi*R^2)

m = 0.056 kg

part c )

Fnet = B-mg = ma

B = rho_air*V*g

B = 42.84 N

a = (B - mg)/m = 755.26 m/s^2

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