How Much Energy Must a Satellite Release to Enter Jupiter's Orbit?

[gpax42]

Tricky Question --> gravitational pull problem

Homework Statement

NASA sends up a satellite that just escapes Earth's gravitational pull. If the satellite is sent to Jupiter (Mass = 2*10^7kg Radius = 70,000km) how much energy must be released by the satellite such that it will successfully enter a stable circular orbit at an altitude of 10,000km from Jupiter's surface? Keep in terms of satellite's mass

Homework Equations

G=6.67*10^-11

(KEf-KEi)+(PEf-PEi)=0

PE= -(GMm)/r

The Attempt at a Solution

PEi= -(GMm)/r1 = -(6.67E-11*2E27)/70000 = -1.906E9*m
PEf= -(GMm)/r2 = -(6.67E-11*2E27)/80000 = -1.334E10*m

KEi=0 --> reference point
KEf=(1/2)*m($$\Delta$$v)^2

(-1.334E10*m)-(-1.906E9*m) = $$\Delta$$PE = -1.143E10*m = $$\Delta$$KE

$$\Delta$$PE=$$\Delta$$KE
1.143E10*m=(1/2)m($$\Delta$$v)^2

$$\Delta$$v^2=2*(1.14E10)
$$\Delta$$v^2=2.287E10 m/s
$$\Delta$$v=1.51E5 m/s

I believe that the satellite must release 1.143E10*m J in the form of increased velocity (KE, 1.51E5 m/s), to maintain that orbit. So it must release energy to increase thrust (velocity)

... Not sure what made the professor take off points. His Comment was "what about V"
does anyone know how I could solve for V?

 

[clamtrox]

You are missing a key piece of information in the assignment: you need to remember that Earth rotates around the sun (at a rate of ~30km/s). In order to send a satellite to jupiter, you must launch the rocket to the direction of Earth's movement, giving it a much larger velocity with respect to Jupiter (which you probably cannot even solve, since Jupiter is moving as well). I really have no clue how you should have been able to solve the problem at all :)

 

[Coreyh7988]

Hello, Yea you need that velocity! I am uncertain if this is right since the title "tricky" makes me feel like I am wrong anyway i think what the issue at hand is...is that you have KEi as zero. Which is not true. The satellite first needed to leave the Earth's gravitational pull. So you have to figure out the velocity necessary for that. which can be found using the eq. v=the sqrt of (2GM/r) where M and r are the mass and radius of the earth. Anyway so the satellite initially has that kinetic energy. I also think you may ignore any initial potential energy because we are soooo far from Jupiter when we escape from the Earth's field. This is the confusing part so there is an equation you can use to find the velocity needed to put a mass in orbit around a larger mass. Its v=sqrt(GM/r) where M is the mass of Jupiter and r is the radius of the path of the satelitte. use that v to find the new kinetic energy and Potential energy you have the rest of the concept correct, Just do KEi+PEi..where PEi=0...ull get some number. then do KEf+PEf. then do...initial energy-final energy to get the amount of energy dropped. I hope that clears some things up.

 

[gpax42]

I think my problem was that I was setting (KEf-KEi)+(PEf-PEi)=0 ... really it was supposed to equal the work of the space shuttle

PEi can be assumed to be zero, and KEi can also be set to zero because the satellite is just escaping Earth's gravitational pull.

Because the satellite is entering an orbit, Centripetal acceleration and Force of Gravity can be set equal to one another

(m*v^2)/R = GMm/R^2

Solve for v^2 and then enter that value into:

(1/2)*m(v)^2-GMm/R = W

I was then able to solve for the amount of energy released...

Thanks for all the helpful advice =)