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# Physics/Satellite's In Orbit

Question
Hi,

I'm preparing for a Physics exam next week and there's a question I simply cannot figure out how to answer:

A satellite leaves the Earth at a speed of 4.00x10^4 m/s. What is the satellite's speed when it is very far from the Earth? G = 6.67X10^-11 Nm^2/kg^2, R(Earth) = 6.37x10^6 m, M((Earth) = 5.98x10^24 kg.

I have tried every formula I can think of but cannot get the answer at the back of the book which is 3.92x10^4 m/s.

I would be ever so grateful for some help with this. Thanks in advance

This problem is an energy conservation problem. When the rocket is launched it has two types of energy; kinetic and potential. When the rocket gets very far away from the Earth it will only have kinetic energy. (Since for gravitational purposes zero potential energy occurs at infinity - aka "very far away").
Making the total energy on launch equal to the total energy far away becomes:
GPEo+KEo=KEf
-G*m1*m2/Ro+1/2*m2*Vo^2=1/2*m2*Vf^2
Which then becomes:
-G*m1/Ro+1/2*Vo^2=1/2*Vf^2
Solving for the velocity far away Vf:
Vf=sqrt[2*(-G*m1/Ro+1/2*Vo^2)]=sqrt[2*(6.67*10^(-11)*5.98*10^24/6.37*10^6)]=3.84*10^4m/s

PS Your answer in the back of the book seems to be a bit off!

Physics