Is there a way which isn't physics related, but just using pure maths to find the solution to the following problem:
If i have two lines of different lengths at t=0 overlapping each other. They are both fixed at a central point and will rotate at different speeds so they're bound to overlap at some point. Knowing the length of these two lines and the respective velocities, how can I find the distance or the time at which the angle between the two is zero once again (so they overlap once again). I've tried circular motion and related rates of change but can't figure out how to do it...
Ok, suppose the two planets rotate with angular velocities $\omega_1$ and $\omega_2$. Measure angles from the direction at time $t = 0$ used as the zero. Then at time $t$ the two planets are at positions $\omega_1t$ and $\omega_2t$. These two are equal iff
$$\omega_1 t - \omega_2 t = 2n\pi, \ \ \ \hbox{ for some integer } n$$
Now you should be able to find all such times $t$.