Launch Ball Highest: Velocity Lost in Arc

[Duhvelopment]

Homework Statement

We're assigned to build a ramp that launches our ball as high as possible. For now we're not taking any friction into account.
Max height: 120cm
Image:

http://i.imgur.com/4DmX0mZ.png

Homework Equations

v = g * t
Ep = m x* g * h
Ek = (1/2) * m * v^2

The Attempt at a Solution

We decided that the best way to approach this would be to construct a cane like shape, where the ball wouldn't be exposed to friction and would be essentially in a "free-fall". After that the ball will enter a half-circle like shape, where-after it will launch back into the air under an angle of 90 degrees. We're trying to find the best possible radius of our arc. When we make our arc too small, too much energy will be lost in the movement translation, and when we make the arc too big, the ball will experience a lot more friction so it won't reach the maximum height. We're not asking for exact solutions but more like guidelines and formulas we could use to calculate the equilibrium.

 

[haruspex]

Please explain more your concern about what happens when the arc is small.
Also, when the ball is near the launch point, do you expect it to be rolling or sliding?

 

[Duhvelopment]

Please explain more your concern about what happens when the arc is small.
Also, when the ball is near the launch point, do you expect it to be rolling or sliding?

When the arc is too small, the angle will differentiate at a much faster rate which will result in a bigger energy loss. We're going to try and keep the friction as small as possible by covering our ramp in a non-friction product. We're a bit unsure whether the ball would be rolling or sliding, because there might be enough friction for it to start rolling, but for the sake of simplicity we're assuming that the ball will be sliding.

 

[haruspex]

When the arc is too small, the angle will differentiate at a much faster rate which will result in a bigger energy loss. We're going to try and keep the friction as small as possible by covering our ramp in a non-friction product. We're a bit unsure whether the ball would be rolling or sliding, because there might be enough friction for it to start rolling, but for the sake of simplicity we're assuming that the ball will be sliding.

I don't see why a tighter curve leads to a greater energy loss. Equations? The key is how smooth the curve is.
If the surface is slippery (and short) enough that it does not achieve rolling, that will definitely help.

 

[Duhvelopment]

I don't see why a tighter curve leads to a greater energy loss. Equations? The key is how smooth the curve is.
If the surface is slippery (and short) enough that it does not achieve rolling, that will definitely help.

Thanks for the reply, I'll get to the equations tomorrow as it's quite late at the moment.

 

[haruspex]

The key is how smooth the curve is.

To expand on that... you have the down ramp as vertical. You don't want to risk the ball hitting the U bend at some angle, so maybe make it not quite vertical.