How Do I Convert v^2/r Into v?

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Homework Statement



Can someone please walk me through the carrying to get the value of v? I am really lost when it comes to this..

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The Attempt at a Solution

 
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  • Handwritten derivation on paper of a satellite's orbital speed: a_s = F_net/m_s, then v^2/2R_E = G m_s M_E / (2R_E)^2 / m_s with m_s cancelled, leading to v = sqrt(G M_E / 2R_E). A sketch shows a circle of radius R_E with an orbit of radius 2R_E and a force F_g arrow.
    Screenshot (81).jpg
    31.3 KB · Views: 488
  • Handwritten derivation on white paper of a banked-curve speed: FN cos θ = mg, FN = mg/cos θ, v²/R = FN sin θ/m, substituted to v²/R = (mg/cos θ) sin θ/m, ending with v = √(Rg tan θ).
    Screenshot (82).jpg
    38 KB · Views: 389
  • Handwritten derivation on white paper of conical pendulum equations: Tcosθ = mg, then Tsinθ/m = v²/R, substituted with mg/cosθ to give v = √(Rg tanθ), with a partial diagram of a circular path and a mg arrow above.
    Screenshot (83).jpg
    34.1 KB · Views: 359

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