Question

Find the speed of a satellite in a circular orbit around the Earth with a radius 3.57 times the mean radius of the Earth. (Radius of Earth = 6.37×10^3 km, mass of Earth = 5.98×10^24 kg, G = 6.67×10^-11 Nm^2/kg^2.)

Answer 1

In a circular orbit with a radius that is 3.57 times the mean radius of the Earth, a satellite moves at a speed of 132.43km/s.

In order to get the solution, we must understand the satellite's planetary motion equation.

What is the satellite's orbital motion equation?

• The earth's mass M and the satellite's mass M are attracted to one another by gravity.

                          $F_g=\frac{GMm}{r^2}$

• The term used to describe centripetal force of,

                         $F_c=\frac{MV^2}{r}$

• When a satellite orbits the earth, these two forces are equivalent. Thus, the velocity will be,

                          $\frac{GMm}{r^2}=\frac{mV^2}{r}\\V=\sqrt{\frac{GM}{r} } =\sqrt{\frac{6.67*10{-11}*5.98*10^{24}}{22.74*10^3} } \\V=132.43*10^3m/s$

As a result, we may estimate that the satellite will move at a speed of 132.43 km/s.

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Answer 2

Answer: The speed of a satellite in a circular orbit around the Earth with a radius 3.57 times the mean radius of the Earth is 4188.11 m/s.

Explanation: To find the answer, we need to know about the equation of motion of a satellite around earth.

What is the equation of motion of a satellite around earth?

• We have gravitational force of attraction between the satellite of mass m and earth of mass M as,

                 $F_g=\frac{GMm}{r^2}$

• The expression for centripetal force of,

                 $F_c=\frac{mv^2}{r}$

• These two forces are equal for a satellite around earth.

                    $\frac{GMm}{r^2} =\frac{mv^2}{r} \\thus,\\v=\sqrt{\frac{GM}{r} }$

How to solve the problem?

• Given that,

                  $r=3.57 R_E=3.57*6.37*10^3=22.74*10^3 km\\M=5.98*10^24kg\\G=6.67*10^{-11}Nm^2/kg^2$

• Thus, the speed of the satellite will be,

                  $v=\sqrt{\frac{6.67*10^{-11}*5.98*10^{24}}{22.74*10^6m} } =4188.11 m/s$

Thus, we can conclude that the speed of satellite will be 4188.11 m/s.

Learn more about motion of satellites here:

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