Sorry to jump in but If I could just ask a couple of questions...
I got the answer to this problem but I dont really understand why it works.
I watched the video
https://www.youtube.com/watch?v=_x0hSuZi8Jsand I get how the relative parallel velocity is kv - vsin(theta)
but he says "only the parallel component changes the distance of the two boats" which made sense but after thinking about it I dont really get it?
In this diagram I split the velocity into parallel and perpendicular components

and it should imply that v⊥
should move the smuggler ship from the black dot to the red dot, so increasing its distance from the coastguard ship but it doesnt have any effect on the distance?
My question basically is how can we ignore v⊥?
I also dont get the first integral he set up (I understand the other 2)
Int v_rel dt = a
I get how Int kvcos(theta) dt = a
but that doesnt seem to be important - how does the relative velocity (that too one which neglects v⊥) have a distance equal to a?
Any explanations greatly appreciated.