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MATHEMATICS - CASE STUDY

    Introduction

 


Launch an Satellite
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NASA is planning to launch an unmanned satellite into a circular orbit at an elevation of 650 km. Engineers need to know the work required to launch the satellite into its orbit in the vertical direction.

What is known:

  • Newton's Law of Gravitation states that two bodies with mass given as m1and m2 attract each other with a force

    FG = Gm1m2/r2

    where r is the distance between the bodies ( from their centers) and G is the gravitational constant, G = 6.6 7x 10-11 N-m2/kg2.
  • The mass of the earth is 5.98 x 1024 kg and is assumed to be concentrated at its center. The average radius of the earth is 6.37 x 106 m.
  • The mass of the satellite is 173 kg and the satellite is launched to an orbit 650 km above the surface of the earth.
     
    Questions

    How much work required to launch this satellite vertically to its orbit?
     
    Approach


 
  • The vertical thrust force needed to move the satellite up equals the gravitational force in magnitude, but points to the opposite direction.
  • Consider a small interval between the earth's surface and the orbit of the satellite. Find the work when the satellite moves through this small interval.