You are part of a design team for future exploration of the planet Mars, where g

Question

You are part of a design team for future exploration of the planet Mars, where g= 3.7m/s2. An explorer is to step out of a survey vehicle traveling horizontally at 33 m/s when it is 1200 m above the surface and then fall freely for 20 s. At that time, a portable advanced propulsion system (PAPS) is to exert a constant force that will decrease the explorer's speed to zero at the instant she touches the surface. The total mass (explorer, suit, equipment, and PAPS) is 150 kg. Assume the change in mass of the PAPS to be negligible.

Find the horizontal and vertical components of the force the PAPS must exert, and for what interval of time the PAPS must exert it. You can ignore air resistance.

Explanation / Answer

Horizontal speed uh = 33 m/s

Gravitational acceleration g = 3.7 m/s/s

Time of descent   t = 20s

Height descended h = 0.5 gt^2 = 0.5x3.7x20^2 = 740m

Height to descend   d = 1200 – 740 = 460m

Vertical velocity after 20s   uv = 3.7x20 = 74 m/s

After 20 seconds the explores will have

Vertical velocity   uv = 74 m/s

Horizontal velocity uh = 33 m/s

She will have vertical acceleration due to gravity g = 3.7 m/s/s

Let us assume the force due to PAPS system will cause ah and av horizontal and vertical retardations so that the speed at the time of landing will be 0

Final vertical velocity =0

Initial vertical velocity = 74 m/s

Retardation = av-g

Height to descend d= 460 m

uv^2 = 2(av-g)v*d

av-g= uv^2/2d = 74^2/2*460 = 5.95 or av = 5.95+3.7 = 9.65

if t is the time of descent then

uv =(av-g)t or t = 74/5.95 = 12.43 s

vertical velocity will become 0 in 12.43 s

horizontal velocity shall become zero in 12.43 s

uh = ah*12.43 , ah= 33/12.43 = 2.65m/s/s

mass of the exporer m = 150 kg

Horizontal component of the PAPS force Fh = 150*2.65 = 398 N

Vertical component of the force Fv = 150*9.65 = 1447.5 N,

time interval t = 12.43 s

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