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Mechanics, rewrite and solve equation, change of variables.

  1. Aug 20, 2013 #1
    Equation found in previous question is #: x''=gsin(a)-b(v^2)
    1. Rewrite the equation # as a differential equation for v as a function of x.
    2. Solve the equation to find v as a function of x.




    Relevant equations: v=x'=dx/dt, x''=v'=a=dv/dt



    Attempt at 1: Using the relevant equations you can rewrite # as
    *: dv/dt=gsin(a) - b((x')^2)
    Which i am scepticle of being correct due to my inability to solve it in the 2nd part.
    Attempts at 2:Tried integrating but the whole((x')^2) confused me, i was thinking of making that (dx/dt)*(dx/dt) but got stuck again.
    Tried to rewrite * but i was expecting a first order LINEAR equation, so that i could use the integrating factor method, so alas that also didn't work.


    Any help would be appreciated, thankyou.
     
  2. jcsd
  3. Aug 20, 2013 #2

    tiny-tim

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    hi fed_ex! welcome to pf! :smile:

    (try using the X2 button just above the Reply box :wink:)
    correct, but you've gone a bit too far …

    try dv/dt=gsin(a) - b(v2) :wink:
     
  4. Aug 20, 2013 #3

    HallsofIvy

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    You are asked to find a differential equation for v as a function of x so you want derivatives with respect to x, not t.
    By the chain rule, x''= dv/dt= (dv/dx)(dx/dt)= v (dv/dx).

    Replacing x'' in the equation by v (dv/dx) gives the equation you want.
     
  5. Aug 21, 2013 #4
    Ah yes! Then separate and enjoy tasty solution.
    Thanks!
     
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