A satellite moves in a stable circular orbit with speed Vo at a distance R from the center of a planet. For this satellite to move in a stable circular orbit a distance 2R from the center of the planet, the speed of the satellite must be??

Question

I said that F=ma, but m doesn't matter since it's constant. So, a0=a1. a=v^2/r. So V1^2/2R = V0^2/R. I ended up with V1 = V0sqrt(2). But that's not the answer. All the multiple choice answers have sqrt's, 2's, and V0's scattered around, but none are what I have. What did I do wrong??

Answer 1

Keplers third law is a neat way to start

r^3=k T^2

but T= 2PR/V so

r^3=K1 (1/v)^2 where k1 is a constant.

So
V^2*r^3= K1
Vo^2*R^3=K1

so if you double r, that must decrease Vo by sqrt (1/8)= 1/(2sqrt2)

check my thinking.

Answer 2

R = orbit radius
µ = gravitatinal constant of body = GM

Circular velocity at R = Vo = sqrt(µ/R)

Circular velocity at 2R = V1 = sqrt(µ/2R)

V1/Vo = sqrt(µ/2R)/sqrt(µ/R)

V1^2/Vo^2 = (µ/2R)/(µ/R) = R/2R = 1/2

V1/Vo = sqrt(1/2)

V1 = Vosqrt(1/2)

V1 = Vo(1/1.41421) = .7071Vo