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

1. Oct 1, 2008

derelictee

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

I know; my work is a mess.

Last edited: Oct 1, 2008
2. Oct 1, 2008

LowlyPion

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)

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

C is c because V=c at x=0