Gravitation and the Principle of Superposition

[brendan3eb]

Homework Statement

How far from Earth must a space prove be along a line toward the Sun, so that that the Sun's gravitational pull on the probe balances the Earth's pull

Homework Equations

F=Gm1m2/d^2

The Attempt at a Solution

d1=distance from probe to Earth
distance from sun to Earth = 1.50x10^11 m
distance from the sun to probe = 1.5x10^11 m - d1
mass of Earth = 5.98x10^24 kg = m2
mass of sun = 1.99x10^30 kg = m3
m1 = mass of probe

I set the force on the probe from Earth equal to the force on the probe from the sun to get:
(Gm2m1)/d1^2 = (Gm1m3)/(d1^2-(3.0x10^11)d1+2.25x10^22)

When I solve for d1, I do not get the answer which is 2.6x10^5 Km. It could easily be a math error, but the fact that I have to use such tedious calculations makes me wonder if I am doing the problem correctly, especially as the mass of sun, mass of earth, distance from sun to Earth are not given.
(Gm2m1

 

[fliinghier]

what is that equation you are using for the probe-sun distance?

 

[brendan3eb]

just F=Gm1m2/D^2

I plugged in 1.5x10^11 - d1 into D^2

 

[fliinghier]

i see. i would suggest you make sure you solve the equation carefully as to not make arithmetic errors. I'm pretty sure that's all this is.

 

[Saladsamurai]

just F=Gm1m2/D^2

I plugged in 1.5x10^11 - d1 into D^2

I do not believe that you are using the equations properly.

I suggest using R_1 to represent the distance between the Earth and satellite and R_2 to represent the distance between the Sun and the satellite.

You seem to be using 1 au-d1 to represent something here. But 1au is not the distance between the SATELLITE and anything; it is the distance between the Earth and Sun only.

Do you see what I mean? You should end up with two unknowns.

Casey

 

[fliinghier]

he's doing it correctly actually. the distance between the sun and the probe can be expressed as a difference of the earth-sun distance and the earth-probe distance.

 

[brendan3eb]

I'll double-check. Thanks for the help fliinghier :)

 

[Saladsamurai]

he's doing it correctly actually. the distance between the sun and the probe can be expressed as a difference of the earth-sun distance and the earth-probe distance.

Yes I see now. He is using two unknowns, and he eliminated one of them already.

Kudos! It usually takes me two steps to acomplish what you have done in one! I will keep your method in mind for future problems.

Casey