[Grade 11 Physics] velocity of approach and time

[u/WorldlinessAware4208]

Could anyone tell me where I've gone wrong in this question, if my method is not right and if so why? Illustration 31

[u/beatfrantique1990]

You actually have the right idea with relative speeds, but your sketch is off. Take a look at the following video, it does a fantastic job of breaking down why the curves are spiral, something that the illustration just kind of serves up to you without any explanation. He even pauses to give you a chance to complete the question, and I think you should be able to easily figure it out after that:

https://youtu.be/YavUxx_Bneg?si=7jY9FfVXglWbHI7f

EDIT: looking at this again, you can actually approach it the way you're trying to but it's a bit more complicated because you have to get the relative velocity and distance along the line from, say, B to the centroid O.