Homework Help: Bead on wire with constant velocity

1. Sep 22, 2011

treynolds147

1. The problem statement, all variables and given/known data
A bead, under the influence of gravity, slides along a frictionless wire whose height is given by the function y(x). Assume that at position (x,y) = (0,0), the wire is horizontal and the bead passes this point with a given speed v0 to the right. What should the shape of the wire be (that is, what is y as a function of x) so that the horizontal speed remains v0 at all times? One solution is simply y = 0. Find the other.

2. Relevant equations

3. The attempt at a solution
I started out this problem by considering the initial and final energies of the bead, $E_{i} = \frac{1}{2}mv^{2}_{0}$ and $E_{f} = mgy + \frac{1}{2}mv^{2}$. From there, I can find the velocity of the bead at a point on the wire to be $v = \sqrt{v^{2}_{0} - 2gy}$. This is the part where I start to get tripped up. I know that the horizontal component of the velocity (v0) would be equal to $v\cos{\theta}$. I already have an expression for v, but I'm not quite sure how to represent the cosine in terms of variables I already have. Once I have that, I'm pretty sure I'd be in the clear for solving this one. Any help would be appreciated! Thanks.

2. Sep 22, 2011

PeterO

If the wire ended at (0,0), the bead would become a projectile, and the horizontal component of the velocity would remain the same, so perhaps if the wire just followed the appropriate parabolic path down that the projectile would have followed???.