Ratio Problem Dealing with Radius and Centripetal Acceleration

[PeachBanana]

Homework Statement

Two identical satellites orbit the Earth in stable orbits. One satellite orbits with a speed v at a distance r from the center of the earth. The second satellite travels at a speed that is less than v . At what distance from the center of the Earth does the second satellite orbit?

Homework Equations

a = (v)^2/r
F = m/a
F = G (M * m )/ r^2

The Attempt at a Solution

I'm really confused on how to manipulate these equations.

Ultimately, I need to find an equation that relates "r" to "v."
r = a * v^2
So if the second satellite is moving slower doesn't that mean "r" has to be bigger? This makes me think of one of Kepler's Laws.

 

[genericusrnme]

F = m/a

That should read F=ma or a=F/m

What does satellites orbit the Earth in stable orbits tell you about the system?
I'm not sure you understand what you're actually doing when you're working with these equations

 

[PeachBanana]

Wow. That was really careless of me. When it comes to problems like these, I don't understand how to do them well. Stable orbits = equal gravitational force?

 

[genericusrnme]

What should the gravitational force equal?

What does it mean if the orbit is stable?

 

[asteriatic]

Kepler's Third Law states that the square of the orbital period is proportional to the cube of the semi-major axis of the orbit. In simpler terms, the farther the satellite is from the center of the Earth, the longer it takes to complete one orbit. In this case, the second satellite is moving slower, so it must have a larger semi-major axis, which is equivalent to the distance from the center of the Earth. Therefore, the second satellite must be orbiting at a greater distance from the Earth's center.

To solve for the exact distance, we can use the equation for centripetal acceleration: a = (v)^2/r. Since we know the speed of the second satellite is less than the first (v < v), we can set up the following equation:

(v')^2/r' = (v)^2/r

Where v' is the speed of the second satellite, r' is the distance of the second satellite from the center of the Earth, and v and r are the speed and distance of the first satellite, respectively.

Solving for r', we get:

r' = (v')^2 * r / (v)^2

This means that the distance of the second satellite from the center of the Earth is directly proportional to the square of its speed. So if the speed is half of the first satellite (v' = v/2), then the distance must be twice as far from the center of the Earth (r' = 4r).

In conclusion, the second satellite must orbit at a greater distance from the center of the Earth in order to maintain a slower speed. The exact distance can be calculated using the equation r' = (v')^2 * r / (v)^2.