Calculating Kinetic Energy for a Satellite Launch

[alicia.kostka]

Homework Statement

So I just finished a physics test and I'm not sure if I approached a question correctly... Our professor gave us the mass of a satellite, and the mass of an imaginary planet and its radius. Then he gave us the amount of initial kinetic energy given to the satellite when launched. His question was "how much kinetic energy will the satellite have when its very very very far away?" (Assuming no other planet or star is exerting a gravitational force on it)

The Attempt at a Solution

The approach I took was as follows... I used the equation for escape velocity v=$$\sqrt{}$$2GM/r ...then using the necessary escape velocity, I used 1/2mv^2 I found the minimum initial kinetic energy for the satellite to escape the planet's orbit. Finally, I subtracted this minimum energy from the actual initial energy given to the satellite to get the final answer.

Does this sound right?

 

[tiny-tim]

welcome to pf!

hi alicia! welcome to pf!

(have a square-root: √ and try using the X² tag just above the Reply box )

yes, that's correct

i think you're confused because you're using escape velocity instead of going back to the basics of KE and PE …

escape velocity is the speed (!) v_e needed to reach r = ∞ at speed zero

so, since PE_∞ is defined as zero, KE_∞ + PE_∞ = KE_r + PE_r, ie 0 + 0 = 1/2 mv_e² + PE_r,

and if v₀ > v_e, then KE_∞ = KE_r + PE_r = KE_r - 1/2 mv_e²

 

[alicia.kostka]

Thanks! I think that's ultimately what I did...I ended up subtracting 1/2mv_e² from the initial kinetic energy given to the satellite when it was on the surface of the planet. I just didn't do it very elegantly. Sometimes I have to play around with equations before I know what I want to do!

 

[tiny-tim]

hi alicia!

(just got up :zzz: …)

that's fine … playing around is often a good way of solving things, but (if you have time in the exams) always try to squeeze out the extra couple of marks by making it elegant!

think "elegant, not elephant!" ​

 

[rwisz]

Yes, your approach is correct. The key concept here is conservation of energy. The initial kinetic energy given to the satellite is equal to the sum of its potential energy and kinetic energy at any point in its orbit. As the satellite moves away from the planet, its potential energy decreases while its kinetic energy increases. At a very far distance, the potential energy becomes negligible and the kinetic energy becomes equal to the initial kinetic energy given to the satellite. So, your approach of using the equation for escape velocity to find the minimum initial kinetic energy and then subtracting it from the actual initial energy given is a valid way to calculate the final kinetic energy. Good job!