physics (sorry about all of these!)

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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??

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??

  • physics (sorry about all of these!) -

    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.

  • physics (sorry about all of these!) -

    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

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