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Homework Help: Calculus: Given parabola and vx in terms of x and c, prove constant velocity

  1. Oct 1, 2008 #1
    Here an AP Physics problem that's really bugging me.
    1. The problem statement, all variables and given/known data

    A particle moves along the parabola with equation y = .5x^2

    part a) I believe I did this correct.

    part b) Suppose that the particle moves with a velocity whose x-component is given by vx = c / (1 + x^2)^.5 Show that the particle's speed is constant.

    Below I have images of the question and my attempted work. I think maybe for the first half of my work I was in the right direction; I got y in terms of t, and I was going to find the derivative to show that there is no acceleration, but I couldn't get the equation to equal y, and I ultimately became confused and went off track.


    3. The attempt at a solution
    http://img363.imageshack.us/img363/7313/scanqa9.th.jpg [Broken]http://g.imageshack.us/thpix.php [Broken]
    http://img218.imageshack.us/img218/2346/scan0001fu2.th.jpg [Broken]http://g.imageshack.us/thpix.php [Broken]
    I know; my work is a mess.
     
    Last edited by a moderator: May 3, 2017
  2. jcsd
  3. Oct 1, 2008 #2

    LowlyPion

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    Homework Helper

    Have you considered that if Vx = c/(1+x2)1/2 that x would be given by the integral?

    In this regard isn't the x position as a function of time given by the integral of Vx(t)

    [tex] \int \frac{c*dx}{(x^2 + 1)^{1/2}} = c*arcsinh(x) + C = c*arcsinh(x) + c[/tex]

    C is c because V=c at x=0
     
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