Question on movement speeds relative to an observer.

Ralphonsicus

Say I am at point X. My friend, Gary, is 20m from X, and my friend Liam is 40m from X. Gary is travelling at 5 metres per second in a circle around me, remaining 20m from X. What speed must Liam travel at to appear, to me, to be travelling at the same speed as Gary? Is it 10 metres per second or is it more complicated?

ghwellsjr

5 metres per second.

Ralphonsicus

Maybe I phrased it wrong. Imagine that Liam wants to stay hidden behind Gary as Gary walks, while Liam himself remains 40m away, and Gary 20m?

ghwellsjr

10 metres per second. But Liam will have to start walking a little before Gary in order to stay hidden.

bahamagreen

It's 10m/s, Liam is going to have to run as fast as Usain Bolt.

Gary's circle of radius 20m has a circumference of pi*D, where D is diameter = 2*20m
So, Gary's circle is about 125.6m around. If he walks 5m of that in one second, then the time around is 125.6m/(5m/1s)=125.6ms/5m=25.12s
Liam would need to walk his circle in the same time as 25.12s
The circumference of Liam's circle is pi*D where D is 2*40m, or 251.2m
Liam needs to go 251.2m in 25.12s, so 251.2m/25.12s=10m/s

George, unless Gary is very thin and Liam pretty fat (for a thought experiment both might be considered identical sized spheres), Liam should subtend a small enough area behind Gary that his image will stay well within that subtended by Gary with plenty of overlap to account for the delaying of his light image, at least at these distances.
But I guess the next step you are anticipating would be to determine the sizes of Gary and Liam, and the respective distances at which the geometrical subtended overlap is overcome by the delay in light. That calculation would be complex because the segment of arc subtended by Gary or Liam would intersect them (if they were spheres with their centers located on the circumference of their walking circles) not as a diameter of their sphere, but as less than their diameters (a little closer to the central observer)... and by then there might be numerous relativistic principles to consider.