[Molten]

Please read last paragraph first.

First of all, I'm going to assume your just working with radial motion.(your kicker only moves about an axis without shifting along it)

Now, the equivalent of position is angle. Instead of measuring in feet or such, you will be measuring this in rotations or radians. This makes the equivalent of linear velocity to be angular velocity(often symbolized in formulas as omegaω). It can be measured in rpm or radians per second. The equivalent of linear acceleration is angular acceleration(often symbolized in formulas as alphaα).

Ok, your sort of right on your idea about x and y. However, your trying to work in Cartesian coordinate system(and if you are going to you would have to consider this and it would effect formulas). I'd work in the polar coordinate system. Instead of using formulas with x and y components, you will be using radius and angle components. Assuming your radius is constant we can get all of our formulas into angle components.

Another thing your going to need to understand is Rotational inertia. Its sort of like mass in your equations.(take this loosely, they are two completely different entities...just happen to be placed the same in formulas sometimes)

To summarize:
position=>angle
velocity=>angular velocity
acceleration=>angular acceleration
mass=>rotational inertia
force=>torque

Example:
Force=mass*acceleration
becomes
Torque=rotational inertia*angular acceleration

Now, I've covered all the basics you should need to understand the steps to find what you want.

First step is to find your rotational inertia. This can be a little confusing to do and I really can't tell you how to do it without knowing the geometry of your kicker. I'd presume that many kickers can be solved for using a rod rotating about end, a ball, and the parallel axis theorem. This is going to get messy pretty fast and you might have to look up some videos to understand rotational inertia and why my math works.

I'm assuming your kicker consists of a stick with a weight at the end. I'm sure your design isn't this simple, but with physics we choose to simplify things to make the math possible. The formula for the stick is simple.

Use: I=(mL^2)/3
m is the mass of the rod, include anything that is distributed between the axis and the end. L is the length of this rod.

Now, for the weight at the end we will assume it to be a ball. However, the formula is complicated because we are not rotating the ball about its center. Lets find the inertia due to its shape.

For a ball about its center use: I=(2mr^2)/5
m is the weight of the ball, include anything that is only at the end and wasn't included in the rod. r is the radius, your really going to have to approximate this one.(it won't be too big of a deal on your final answer, but it is good to be close)

Now for the parallel axis theorem. This theorem explains that a mass rotated about its center will have less inertia then the same mass rotated about a different axis. This formula helps us to calculate the additional inertia.

For the additional inertia use: I=m*d^2
m is the mass of the ball because that is what has been displaced. d is the distance of displacement. In this case d should be the L used in the above formula for the rod.

Now, to find the total rotational inertia just add the three inertias we calculated above and you will have your final inertia.

My main confusion is on what your wanting to find. You could find the torque required to get certain speeds of rotation. You could find the energy within the kicker which might relate to the distance you get. You could find the angular momentum which could also be used to find distance. Please explain the end goal a bit clearer and send me a PM so I'll be sure to see it. I'll gladly come back and go in more depth on parts I might have overlooked or to continue. Also, if you are having trouble putting in your data into the above formulas, just give me the info and I'll help you figure it out.

Sorry I couldn't finish the problem,
Jason