A Mars-lander enters the Martian atmosphere much faster than the terminal speed

Question

A Mars-lander enters the Martian atmosphere much faster than the terminal speed and is designed to land safely if it can slow to constant vertical speed of 2 m/s after deploying a parachute with an area of 1000 m2 . Its drag coefficient is 186. The surface gravity on Mars is 3.71 m s2 , and the density of the atmosphere near the surface is 0.01 kg m3 .

a. What is the maximum mass of the Mars lander to ensure it can land safely?

b. The mission designers consider a smaller lander with a mass of 480 kg. What is the required size of the parachute now?

c. To test the lander on Earth, where g = 9.8 m/s2 and the atmospheric density is 1.0 kg m3 , how big should the parachute be for the terminal speed to be 2 m/s, if the mass of the lander is 480 kg.

Explanation / Answer

part a:

let maximum mass be m kg

to safely land, drag force should balance the weight of the lander.

then 0.5*density*speed^2*drag coefficient*surface area=mass*acceleration due to gravity

==>0.5*0.01*2^2*186*1000=m*3.71

==>m=0.5*0.01*2^2*186*1000/3.71=1002.7 kg

hence required mass of the lander is 1002.7 kg

part b:

area of the parachute=2*mass*acceleration due to gravity/(density*speed^2*drag coefficient)

=2*480*3.71/(0.01*2^2*186)=478.71 m^2

part c:

area=2*mass*acceleration due to gravity/(density*speed^2*drag coefficient)

=2*480*9.8/(1*2^2*186)

=12.645 m^2

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