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Originally Posted by Biff
That is assuming all the rim velocity from the wheel(s) are transfered to the ball, As I understand the design the ball is only driven on one side. If you have wheels on both sides of the ball then you are more likely to have a closer match between wheel speed and ball speed, There is still going to be some slippage unless you have a very long contact surface. IE the belt idea.
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Just to expand on Biff's ideas, I think with a wheel you are going to have to measure velocity rather than calclulate it. Look at it this way, if you have a single long belt and the ball is pressing against something to provide traction against the belt, the ball will come out of the launcher at half the speed of the belt (but spinning rapidly). You can mock this up easily -- put a tennis ball on a table and move it along by pressing down with a book on top of it. The tennis ball will always come out because it is going slower than the book. With twin belts completely gripping the ball, it will exit at a speed that is the average of the two belts.
A wheel launcher is more problematic. In a single-wheel launcher, the maximum speed that ball can be released at is half the speed of the wheel at its rim. Because of slippage -- something that we assumed was zero with the belt launcher -- the bal won't really come out at that speed. It will be slightly (or a lot) slower. A 2-wheel launcher has a theoretical maximum speed of the speed of the wheel at its rim.
Generally speaking, if you are trying to get maximum ball velocity with a minimum of energy used, you are using either a 2-wheel (or, more properly, two or more wheels on each side of the ball) or 2-belt launcher. Unless you are using a pneumatic, trebuchet, spring, or cold fusion launcher.