A planet is in circular orbit around a star. The period and radius of the orbit

Question

A planet is in circular orbit around a star. The period and radius of the orbit are T = 4.3*10^7s and R = 2.34*10^11 m, respectively.
The Earths mass and radius are ME = 5.97*10^24 kg and RE = 6.378 *10^6 m, respectively. A satellite moves in a circular orbit around the Earth with speed v = 3.07 km/s.

(a) What is the minimum energy required to launch a probe of mass m = 120 kg from Earth into outer space?

(b) Calculate the speed which the probe must have at the surface of the Earth in order to achieve this.

(c) Note that the velocity you just calculated is called the escape velocity, and it is the same for any object, regardless of mass. Prove that this is the case!


Please show in clear steps with correct formulas in order to receive all points. Thank you!!!!

Explanation / Answer

a)
E=KE=(1/2)mv2

velocity will be the escape velocity so

E=(1/2)m(2GM/r)

E=.5(120)(2*6.673x10-11*5.9742x1024/6378100)

E=670832 joules

b)escape velocity is

v=(2GM/r)=(2*6.673x10-11*5.9742x1024/6378100)

v=11180m/s

c) escape velocity only depends on the mass of the planet, not the mass of the object. you can se this because there is no m term in the equation.

Hope that helps

(I didn't realize this problem was to figure out escape velocity on your own when I was giong through it, but to derive you say that kinetic energy is equalt potential energy of gravity and solve for v:

(1/2)mv2=GMm/r

then you get the v=(2GM/r) that I used)

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