Calculating minimum energy to remove satellite from orbit

AI Thread Summary
The discussion revolves around calculating the speed of a satellite and the minimum energy required to remove it from orbit. The satellite's speed is determined to be approximately 7272.88 m/s using gravitational equations. The minimum energy needed to move the satellite far from Earth is calculated as 2.38e11 J, but confusion arises regarding the correct method to account for the satellite's existing kinetic energy. Clarification is provided that the kinetic energy should be halved in the calculations, leading to the realization of an elementary mistake in the energy calculation. The conversation emphasizes the importance of correctly applying energy conservation principles in orbital mechanics.
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Homework Statement


A satellite of mass 4500 kg orbits the Earth in a circular orbit of radius of 7.6 x 10^6 m (this is above the Earth's atmosphere).The mass of the Earth is 6.0 x 10^24 kg.
What is the speed of the satellite?

What is the minimum amount of energy required to move the satellite from this orbit to a location very far away from the Earth?



Homework Equations


F = mSatellite*a = G*mEarth*mSatellite/r²
a = G*mEarth/r² = v²/r -> v=SQRT(G*mEarth/r)
r = 7.6 x 10^6 km
mEarth = 6 × 10^24 kg

v = 7272.88 m/s

minimum amount of energy = increase of potential energy = 0 - (-GMm/r) = GMm/r = 6.67*10^(-11)*6.0*10^24*4500/7.6*10^6 = 2.38e11 J


The Attempt at a Solution


I know that I obtained the correct velocity, but when I try to solve for the energy required I get the wrong answer. Is the above equation for calculating the minimum amount of energy incorrect? What should I be doing?
Thanks in advance!
 
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Escaping the Earth (to very far away) means having kinetic energy equal to GMm/r.
You have some KE already, so you need GMM/r - the present KE.
 
Ahhhh I see, so I would need to do something such as

2.38e11 J - (4500 kg)(7272.88 m/s)^2 =-209,935 J

But I cannot have a negative number for this, as energy must be put into remove the satellite from the Earths orbit.
If I am correct, what I did above was total energy - energy of satellite... Shouldn't this be correct, since the energy needed to remove the satellite plus the energy of the satellite must equal total energy.
I feel that I am making an elementary mistake, so could someone please help clarify this for me?
 
You forgot to divide (4500 kg)(7272.88 m/s)^2 by 2!

It looks like we're working on the same homework questions:)
 
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