Sliding on Sphere with Friction

[buttermellow7]

Homework Statement

Snowboarders, standing at the top of a spherical mountain, hold a contest to see who can slide farthest down the mountain before taking flight. Contestants must decide whether to use High Friction Wax A or Low Friction Wax B. "M-dog" Matt Curcio was one of the contestants, and, as a recent graduate of this department, made the correct choice. Which did he choose, and why?

Homework Equations

$$1/2mv^2-\int Fdx=mgh$$
(would we need to take into account rotational kinetic energy? In this case, the factor of 1/2 would disappear.)
$$F_\mu=\mu_kF_N$$
$$\sum F=ma=mv^2/R$$
$$h=R\cos(\theta)$$

The Attempt at a Solution

I think qualitatively you would want the high friction wax, because then more of your potential energy would be lost due to friction and thus not converted into kinetic energy. You would be going more slowly, so you wouldn't fly off the mountain as quickly (this last part I only feel instinctively is true; I'm having trouble proving it). A higher coefficient of friction would also mean a smaller normal force, and thus less outward force on the snowboarder.

However, when I try and do the math to prove it, I think I get a little lost. I know that we're looking for the place where the normal force is 0. So I try
$$F_\mu+F_g+F_N=mv^2/R$$
$$(\mu_k+1)F_N+mg\cos\theta=mv^2/R$$
$$0=\frac{(mv^2/R-mg\cos\theta)}{(\mu_k+1)}$$
So I really don't know what to do with the velocity factor. Can anyone help me work this all the way through? Am I even approaching it the right way? I'm supposed to be able to do this question in under nine minutes, so I think there's probably a quicker way.

 

[zorro]

This might help:
https://www.physicsforums.com/showthread.php?t=451082&highlight=uniform+rod