You are designing a spacecraft intended to monitor a human expedition to Mars (mass 6.42×1023kg, radius 3.39×106m). This spacecr

Question

1 month ago

12

You are designing a spacecraft intended to monitor a human expedition to Mars (mass 6.42×1023kg, radius 3.39×106m). This spacecr

aft will orbit around the Martian equator with an orbital period of 24.66 h, the same as the rotation period of Mars, so that it will always be above the same point on the equator. What must be the radius of the orbit?

Physics

1 answer:

Softa [3K] · Answer 1 · 2026-03-28T16:10:40+03:00

The height is h = 17 10⁶ meters above the surface of Mars. To determine this, we apply Newton's second law according to the universal law of gravitation, represented by F = m a. The centripetal acceleration a is expressed as v² / r. Applying the gravitational force we have G m M / r² = m v² / r. Given that the speed of the object remains constant, we derive v from d / t, where d is the circumference and t is the orbital period. Substituting gives us d = 2π r and v = 2π r / T. Replacing these values leads to the equation G M / r² = (4π² r² / T) / r, so r³ = G M T² / 4π². Converting time into SI units, T = 24.66 h converts to 88776 seconds. Ultimately, the computed value of r is 2,045 10⁶ m, and after subtracting Mars’ radius of 3.39 10⁶ m, we find the height h to be 17 10⁶ m.