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/m3 (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. A) What should be the radius of these balloons so they just hover above the surface of Mars? B) What should be the mass of these balloons so they just hover above the surface of Mars? C) If we released one of the balloons from part A on earth, where the atmospheric density is 1.20 kg/m3, what would be its initial acceleration assuming it was the same size as on Mars? D) If on Mars these balloons have five times the radius found in part A, how heavy an instrument package could they carry?

Explanation / Answer

a) Let r be the radius of balloon (spherical).

For just floating,

weight of balloon= weight of displaced air

or, 4r² * 4.9 = (4/3)r³ * 0.0154*1000

or, r= 0.95 m

b) mass of balloon, m= 4r² * 4.9= 4*3.14*0.95²*4.9 = 55.54 g

c) net force on the balloon on earth,

F=buoyant force–weight

= Volume*air density*g–mg

= (4/3)*0.95³ *1.2*9.8–55.54*10–3*9.8

= 41.67 N

acceleration, a= F/m = 41.67/0.055=750.25 m/s

d) here radius, R= 5*0.95= 4.75 m

mass of balloon, m'= 4R² * 4.9= 4*3.14*4.75²*4.9 = 1.4 kg

Let M be the mass of instrument package.

then, M+m' = (4/3)R³ * 0.0154

or, M = (4/3)*4.75³*0.0154–1.4 = 5.5 kg

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