I want to build an automated pinball game.
The plan is to place two light barriers (A and B) on the field. The time from A to B is measured and then the needed time from B to the flipper arm(s) C is calculated.
An idealized environment is assumed. By that I mean that I assume rolling without slipping or sliding. I assume no friction.
$$s = \frac{1}{2} \cdot a \cdot t^2 + v_0 \cdot t + s_0$$ $$t_{BC} = \frac{2 \cdot (s - s_0)}{v_{max} + 2 \cdot v_0}$$ $$t_{BC} = \frac{2 \cdot (s_2)}{v_{max} + 2 \cdot v_0}$$
By the law of conversation of energy I come up with
$$m \cdot g \cdot h = \frac{1}{2} \cdot m \cdot v^2 + \frac{1}{2} / w^2$$
$$v_{max} = \sqrt{g \cdot h \cdot \frac{10}{7}}$$
and
$$v_0 = \frac{s_B - s_A}{t_B - t_A}$$
The idea is to measure the current velocity in point B and use it as $v_0$ to calculate the needed time from B to C.
My questions are pretty simple:
Is my idea correct? I'm mostly uncertain if I'm allowed to assume $v_0$ in the way I'm doing it here.
If not: what would be the better way?
If it's somehow correct: What are possible uncertainities by idealizing the setting in that way?
After getting a few questions:
I prepeared a little drawing and a photo to make it more clear, unfortunately I don't have the reputation to upload any images. Here is a second try: image
The field I'm using is actually using this one: https://www.youtube.com/watch?v=YbeXRZl8hlM
But I use only the building blocks and photo transistors. The "magic" is done using an FPGA.
My worry is if the basic concept is correct at all and if I can calculate the time from B to C in the way I want to do it.