What RPM is needed to launch a ball at velocity X?

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To determine the RPM needed for a ball launcher to achieve a velocity of approximately 18 m/s, the diameter of the wheels (13 cm) and the mass of the ball (0.57 kg) are key factors. The challenge arises from the need to accelerate the ball quickly, which reduces the contact time with the wheels, impacting the required RPM. Calculating the work needed to move the ball over the distance in contact with the wheels is complicated by friction. Understanding the fraction of a wheel revolution while the ball is in contact can help calculate the impulse force on the ball. The discussion emphasizes the importance of maintaining wheel speed to match the tangential speed necessary for launching the ball effectively.
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Imagine a ball launcher which is composed of two rubber wheels that spin in opposite directions such that a ball that is fed in, will be launched from the other side at X m/s.

I am trying to figure out how many RPM the wheels need to be spinning to launch a ball at Velocity X.

Knowns: The required velocity(~18m/s), The diameter of the wheels(13cm), The mass of the ball(0.57kg)

However while thinking about this problem I've run across some issues, such as: If the ball needs to be accelerated to a faster velocity, then the ball stays in contact with the wheels for less time, yet the same distance, however needs to leave the launcher faster. Will this have an effect? I was thinking about working the problem backwards, eg: find the amount of work required to move the ball over "x" distance(the distance it is in contact with the wheels), and then find the # of RPMs required to complete that work, however friction makes that option inaccurate.

Any ideas? Thanks in advance, - Peter
 
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Just a quick question. If you were to feed the ball in without powering the wheels, what fraction of a revolution would the wheels turn while in contact with the ball. If you knew this you could calculate for the impulse of force on the balls.
 
Thread moved from General Math to General Physics. SilverVenom -- is this a project or a homework/coursework assignment? If it's a project, this is the best forum for the question. If it's a homework/coursework assignment, I can move the thread to Homework Help, Intro Physics.
 
If you assume that the speed of the wheels doesn't drop much while the ball is being accelerated, then the speed of the ball is equal to the tangential speed of the wheels.
 

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