I'm trying to calculate the optimal speed (minimizing time) to intercept another object that is moving in the same circular path as me. Here's the catch, there are some constraints:
- There is a mininum value for the time ($t_{min}$).
- My speed has a minimum ($\omega_{1min}$) and a maximum ($\omega_{1max}$) value.
My initial position is $\alpha_1$ and the target's initial position $\alpha_2$.
$$\Delta_0=\alpha_2-\alpha_1$$
I don't care how many "turns" each object does before the encounter. So the final difference is $2{\pi}N$. Where $N$ is any integer number including 0 and negative values. So here is my main equation: $$\Delta=\Delta_0-(\omega_1-\omega_2)t$$ solving for $\omega_1$ and replacing $\Delta$: $$\omega_1=\omega_2+\frac{\Delta_0-2{\pi}N}t$$ Of course my first approach is to solve using $t=t_{min}$ and $N=0$. But what happens if the resulting angular speed is below the lower limit ($\omega_1<\omega_{1min}$) or above the upper limit ($\omega_1>\omega_{1max}$)? Should I increase $N$? Decrease $N$? Use $\omega_{1min}$ or $\omega_{1max}$ and solve for $t$?
There are so many possibilities that I don't know which one I should I apply under each condition. I expect that the solution is some sort of iteration of steps, which is fine for me because this is going to be converted into programming code.
Notes: I don't know the relation between $\omega_2$ and $\omega_{1min/max}$, but I do know all of them are positive:
$$0<\omega_2$$
$$0<\omega_{1min}<\omega_{1max}$$
I also realize that in some cases there may be multiple solutions, in such cases I would choose the solution where $N$ is closer to zero.
