An unmanned spacecraft of mass 492 kg, is in a circular orbit around the earth,

Question

An unmanned spacecraft of mass 492 kg, is in a circular orbit around the earth, observing the earth's surface from an altitude of 53.8 km.
part A. What is the orbital speed of the spacecraft?
part B. What is the period of the spacecraft's orbit?
part C. What is the escape velocity of the spacecraft?
part D. To the dismay of scientists, an electrical fault causes an on-board thruster to fire, decreasing the speed of the spacecraft by 23.9 m/s. If nothing is done to correct its orbit, with what speed will the spacecraft crash into the earth's surface?

Explanation / Answer

When the spacecraft of mass 492kg is orbiting around the earth

at the altitude is h = 53.8km = 5.38*10^4m

the radius of the earth R = 6378.1km = 6.378*10^6 m

the speed of the spcecraft v = 23.9m/s

when the spaceceaft orbiting then the force acting on it is

             F = mv^2/r

   and from gravitational force

            F = GMm/R^2

so   mv^2/r = GMm/R^2

      v^2/r = GM/r

   (v + 23.9)^2/(5.38*10^4 + 6.38*10^6) = (6.67*10^-11)(5.98*10^24) /(6.38*10^6)^2

                                         = 663043600

    v + 23.9 = 7940

   v = 7916 m/s

then the speed v =  28497.6km/h

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