TBR physics question!

[redbird1133]

Hello! I've been struggling on this for ages and was wondering if someone could please try to clarify!

TBR example 2.9a (book 1):
When a planet orbits the sun, a force acts upon it that depends on its orbital radius. How does the resulting tangential speed of the planet relate to the radius, assuming that the orbit is essentially circular?
A. The speed is inversely proportional to the square root of the orbital radius
B. the speed is directly proportional to the orbital radius
C. The speed is inversely proportional to the orbital radius
D. The speed is inversely proportional to the square of the orbital radius

So I'm looking at equation:
F=ma where a= v^2/R
F= m(V^2/R)
Then here, I multiply out F=(mV^2/R), multiply both sides by R and take the square root, those look directly proportional to me but the book gets V proportional to root (1/R)!
Can anyone help? Thank you!

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[Skizye]

What you have done assumes that the force on the planet is the same regardless of the radius of its orbit. Realize that the centripetal force is gravity, so you should set Gmm/(r^2) = mv^2/r. From this equation you can see that A should be correct.
Also, it can help to think about these questions qualitatively. Kepler's Law says that the radius vector of a planet in orbit sweeps out an equal area per unit time regardless of its radius. Thus a planet farther from the sun is moving slower because its radius vector sweeps out a larger area in smaller distance moved by the planet. So your intuition should tell you that the speed should be decreasing as the radius increases and there is no way they can be directly proportional!

 

[redbird1133]

Ah ok, thanks, the qualitative thinking helps a lot.

I'm still slightly confused because the book also uses the F=(mV^2/R) equation and still manages to get inversely proportional.