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 5.30 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) Would it go up or down?

E.)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)Buoyancy force =Weight of air displaced by ballon

pairVg =mg

pair(4/3)pi*R3 = pballon*4piR2

0.0154*(1/3)*pi*R = (5.3*10-3)

R=1.032 m

b)Mass of the ballons

M =pballon*(4pi*R2) =(5.3*10-3)*4pi*1.0322

M=0.071 kg

c) The density of the air on earth, = 1.20 kg/m3  

      The volume of the balloon is

                V = (4/3)()(1.032 m)3

                    = 4.6 m3

      In this case, the net force acting on the balloon is  

                     F = Vg - mg = ma

      Then, the initial acceleration of the balloon is

        a = [Vg - mg ]/m

           = [(1.20 kg/m3)(4.60m3)(9.8 m/s2) - (0.071kg)(9.8 m/s2)]/(0.071 kg)

           = 752 m/s2

  d)acceleration is upwards ,since a>g

e) Given

Rn=5*1.032=5.162 m

=>pair*(4/3)piRn3*g =(Mballon+Mload)g

Mload =pair*(4/3)piRn3-pballon*4pi*Rn2

Mload =0.0154*(4/3)pi*5.1623-(5.3*10-3)*4pi*5.1622

Mload =7.1 Kg

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