Intercept moving target

Question

Here's a problem that has bugging me for a while:

Say I have a friend that is passing my house. My friend is moving at a constant speed in a perfectly straight line. If she already hasn't done so, she will pass my house at a closest approach and then start moving away from my house. I want to meet her, so I leave my house right when I spot her. I also move in a perfectly straight line at a constant speed that is different to hers. Problem is, in what direction should I start walking? If I calculate a lead based on her initial position, her and my speed and her direction of movement and walk in the direction of where the lines intersect, our paths will cross in front of her if she is moving towards my house (because the time it took for me to intersect her path is shorter than the time it would take for me to walk to her initial position), and behind her if she is moving away from my house (because it now takes longer to cross her path than it would take to walk to her initial position).

Is it possible to calculate a perfect solution without using a iterative method?

(Sorry if this has been asked hundred times before).

Answer 1

note that when you walk at an angle you have a lateral speed as you traverse vertical distance and a longitutinal speed as you traverse horizontal distance. The sharper your angle the slower your longitudinal speed is (and the faster your lateral speed). Adjust your angle so that your lateral speed matches hers proportionately and you will cross paths.

If you start walking as she passes your house at a speed of v and you are walking at a faster speed of w. Choose an angle $\theta$ such that $w*\cos \theta = v$. Then your lateral speed matches hers and you will be in lock step vertically and your paths will cross.

Answer 2

The locus of the points that can be arrived at simultaneously by a pursuer and a fugitive is a circle. Bear in mind that the circle's size and position is always in a state of flux.

If the two velocities are equal, then the circle degenerates into the perpendicular-bisector of the line segment connecting the pursuer and fugitive; in this case, the fugitive need merely avoid approaching the line.

If the fugitive's velocity exceeds the pursuer's, then the fugitive is outside the circle and can avoid capture by merely not approaching the circle.

If the pursuer's velocity exceeds that of the fugitive, then the circle surrounds both, and the pursuer can effect capture by proceeding toward the point of that circle that the fugitive's present path will intersect.