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Time to complete a loop

  1. Jul 4, 2011 #1
    Calculate the time t to complete a loop of a rollercoaster of radius R by a car that has initial velocity Vo, with friction u.

    We can calculate the velcity V if we have alpha, the angle that the car is in the rollercoaster

    acceleration in alpha = g.sen alpha - acceleration by atrict


    Fc=N+g.cos alpha


    [itex]A(\alpha ) = g.sen\alpha - (v²/R + g.cos \alpha)u [/itex]


    and
    [itex]V = \sqrt{Vo² + \int a.ds} [/itex]



    The problem is now, I don't know how to solve the equation. I've tried to to this, but I don't know if it's right.



    [itex]A(\alpha ) = g. (sen\alpha +cos \alpha u) -(v²/R) U [/itex]

    [itex]A(\alpha ) = g. (sen\alpha +cos \alpha u) -(v²/R) u [/itex]
    [itex] v² = g. (sen\alpha +cos \alpha u) - A (\alpha) .u/R [/itex]
    [itex] Vo² + \int a.ds =( g. (sen\alpha +cos \alpha u) - A (\alpha)) .u/R [/itex]


    [itex] a.ds = d(Vo²)- d(g. (sen\alpha +cos \alpha u)) .u/R - d(A (\alpha) .u/R [/itex]
    [itex] a.ds = d(Vo²)- d(g. (sen\alpha +cos \alpha u)) .u/R - d(A (\alpha) .u/R) [/itex]


    Now we have a equation with da and ds, alpha = S/R so alpha depends on dS and da depends on a, ok, how do we integrate this?
     
  2. jcsd
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