Closest approach of two particles.

[luckykaa]

So, I'm trying to work out whether a projectile will hit a moving target.  That is, it its closest approach within the collision sphere of the target.

I have the position and velocity of the target and the position and velocity of the target.  The obvious way of doing this is work out the position w.r.t time, use pythagoras to work out the distance (or distance squared because it makes things easier), differentiate to work out the time of closest approach and then put t back into the previous formula to work out the distance. 

It works.  Probably.  But there has to be a nicer way.  It's not like I need to know the time this happens or the actual positions.  Just the distance.  Adding the extra variable just seems to be making things horribly complicated. 

So far what I have it that we can assume that the target is stationary at the origin, and subtract its position and velocity from our projectile, and work out closest approach to the origin.  Now, this will be when the origin is at a right angle to the modified vector.  I just can't work out how to establish how close this is.

Advice, mathematical geniuses of the interwebs?

Comments

[raygungothic.livejournal.com]

Ingaborg beat me to it pretty comprehensively.