Satellite Orbiting Earth Energy

[blackheart]

1. A 962kg satellite orbits the Earth at a constant altitude of 99km. How much energy must be added to the system to move the satellite into a circular orbit with altitude 195km?

2. Homework Equations :

E = (-GMm)/2r
$$\Delta E$$ = E_f - E_i

3. My work (which produced the wrong answer)...

Ei = (-GM_Em_s)/(2(r_E + 99000m))

Ef = (-GM_Em_s)/(2r(_E + 195000m))

$$\Delta E$$ = ((-GM_Em_s)/2)((1/r_f)-(1/r_i)

The answer needs to be in MJ. I got 858MJ which is incorrect.

 

[Delphi51]

Hi blackheart. The method looks good. I got about half your answer.
Did you get 2.26 x 10^-9 for the difference between the 1/r terms?

 

[blackheart]

Oh! I didn't divide by two... I have it now. Thanks a bunch.

 

[Delphi51]

Most welcome.

 

[rwisz]

I would like to clarify that the formula used in the work provided is not accurate for this scenario. The formula used, E = (-GMm)/2r, is for calculating the potential energy of a satellite in an elliptical orbit, not a circular orbit. To accurately calculate the energy required to move the satellite into a circular orbit at a higher altitude, we need to use the formula E = (-GMm)/r, which calculates the potential energy of a satellite in a circular orbit.

Using this formula, we can calculate the initial energy (Ei) of the satellite in its current orbit at 99km altitude to be:

Ei = (-6.67x10^-11 Nm^2/kg^2)(5.97x10^24 kg)(962 kg)/(6.38x10^6 m + 99000 m) = -5.86x10^10 J

Next, we can calculate the final energy (Ef) of the satellite in its desired circular orbit at 195km altitude to be:

Ef = (-6.67x10^-11 Nm^2/kg^2)(5.97x10^24 kg)(962 kg)/(6.38x10^6 m + 195000 m) = -2.95x10^10 J

Therefore, the change in energy (ΔE) required to move the satellite into the circular orbit at 195km altitude would be:

ΔE = Ef - Ei = (-2.95x10^10 J) - (-5.86x10^10 J) = 2.91x10^10 J

Converting this energy into MJ, we get:

ΔE = 2.91x10^10 J x (1 MJ/10^6 J) = 29.1 MJ

Therefore, the amount of energy required to move the satellite into a circular orbit with altitude 195km is approximately 29.1 MJ.