some shooting

i'm at point A in a two dimensional field and there's a furry at B. the furry runs with velocity v, and the bullet from my gun travels with constant speed u. where/what direction should i aim to hit the fucker?

assuming |u| > |v|, it depends on what 'time' you want him to get shot

B+\vec{v}t=A+\vec{u}t
\frac{B-A+\vec{v}t}{|\vec{u}|t} = \vec{x} = \frac{\vec{u}}{|\vec{u}|}

naturally not just any time 't' will work. only those 't' where \vec{x} is a unit vector. and you can see that as 't' gets bigger, the difference in the starting locations starts to matter less and less. \vec{x} is the direction you need to shoot.

this problem reminds me of diophantine equations, although those only deal with integer solutions.

next step would be to find the smallest positive 't' such that this works, or given a time 't_1', what is the next time 't_2' that gives a solution.

if you just want 'any' solution, then you can fix \vec{x} and solve for 't', although solving in this way, 't' is not promised to be positive.

nice problem anon