Gravity equation -- check my work please?

[sp3sp2sp]

Homework Statement

My answer seems really off and wondering if someone could just look at it before I submit answer?

Two 65 kg astronauts leave Earth in a spacecraft , sitting 2.4 m apart
How far are they from the center of the Earth when the gravitational force between them is as strong as the gravitational force of the Earth on one of the astronauts?
Express your answer with the appropriate units.

Homework Equations

F = Gm_1m_2/r^2
G = 6.67e-11N
m_earth = 5.98e24kg

The Attempt at a Solution

1st get gravitation force between the two astronauts then use this value to get distance from center of earth.

F = [6.67e-11N (65kg*65kg)] / (2.4m^2) = 4.89e-8N = force between the astronauts.

r = sqrt [ Gm_1m_2 / F] = 6.67e-11N (65kg*2)(5.98e24kg) / 4.89e-8N
= 1.03e12m = r = distance between Earth and the astronauts.

so my answer = 1.03*10^12m.

thanks in advance if someone could look it over really quick?

 

[BvU]

seems really off

Why ? What did you expect ? something of the order of 10²⁴, of the order of 10⁶ ? or of the order of what you found ?

 

[TSny]

Why the factor of 2 in the following?

r = sqrt [ Gm_1m_2 / F] = 6.67e-11N (65kg*2)(5.98e24kg) / 4.89e-8N

Otherwise, your work looks good to me.

 

[Doc Al]

Your method looks good (once you correct the error that @TSny points out).

Suggestion: Solve it symbolically and only plug in numbers at the last step -- things cancel and the calculation is easier.

 

[haruspex]

Homework Statement

My answer seems really off and wondering if someone could just look at it before I submit answer?

Two 65 kg astronauts leave Earth in a spacecraft , sitting 2.4 m apart
How far are they from the center of the Earth when the gravitational force between them is as strong as the gravitational force of the Earth on one of the astronauts?
Express your answer with the appropriate units.

Homework Equations

F = Gm_1m_2/r^2
G = 6.67e-11N
m_earth = 5.98e24kg

The Attempt at a Solution

1st get gravitation force between the two astronauts then use this value to get distance from center of earth.

F = [6.67e-11N (65kg*65kg)] / (2.4m^2) = 4.89e-8N = force between the astronauts.

r = sqrt [ Gm_1m_2 / F] = 6.67e-11N (65kg*2)(5.98e24kg) / 4.89e-8N
= 1.03e12m = r = distance between Earth and the astronauts.

so my answer = 1.03*10^12m.

thanks in advance if someone could look it over really quick?

This is an example of one of the many reasons it is better to work entirely symbolically, only plugging in numbers at the end. Using m for the masses of the astronauts and x for their separation:
$\frac{Gm^2}{x^2}=F=\frac{GMm}{r^2}$
$mr^2=Mx^2$.
Edit - got delayed in posting, so beaten by Doc Al.

 

[sp3sp2sp]

thanks for the help . I wrote 65kg*2 because m_2 = the mass of the 2 astronauts. That is wrong? thanks

edit; i mean m_2 = mass of the 2 astronauts and m_1 = mass of earth.
Thats why I wrote total mass of the 2 astronauts

1.03*10^12m is wrong answer

 

[haruspex]

thanks for the help . I wrote 65kg*2 because m_2 = the mass of the 2 astronauts. That is wrong? thanks

edit; i mean m_2 = mass of the 2 astronauts and m_1 = mass of earth.
Thats why I wrote total mass of the 2 astronauts

Again, you might have avoided that blunder by using the purely algebraic approach.
Why would you add their masses together? What is being attracted to them as a combined object?

 

[sp3sp2sp]

OK I should have used algebraic before plugging but is the *2 wrong because the question stem asks:

"How far are they from the center of the Earth when the gravitational force between them is as strong as the gravitational force of the Earth on one of the astronauts?"

or am I getting the concept wrong? thanks

 

[sp3sp2sp]

i got 7.28*10^11m after using mass of just the one astronaut.
thanks for any more help.

 

[haruspex]

OK I should have used algebraic before plugging but is the *2 wrong because the question stem asks:

"How far are they from the center of the Earth when the gravitational force between them is as strong as the gravitational force of the Earth on one of the astronauts?"

Yes, that's the right reason. (I misread your equation and thought you were adding them in the case of their attraction to each other.)

i got 7.28*10^11m after using mass of just the one astronaut.
thanks for any more help.

Well, as I showed it should be x√(M/m) = 2.4√(5.98e24/65)= your answer.