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Position, speed and heading are known, and the size and shape of these objects are also known. Gravity is irrelevant, so these objects will be moving in straight lines. I tagged this with geometry and trigonometry but I'm not sure exactly where it should go.

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This sounds very much like a problem in programming (in part, at least) - if so, I would suggest that you do a google search for "collision detection". Here is one page that might be helpful: euclideanspace.com/threed/animation/collisiondetect/index.htm –  Martin Wanvik Apr 3 '12 at 15:40
    
If you didn't already do this, drawing a picture and the relevant data may help. –  john w. Apr 3 '12 at 20:07
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It depends on the shape of those objects. What is it ? Here is an answer assuming they are spheres.

If $x_1(t)$ and $x_2(t)$ are the positions of the centers of both objects at time $t$ (assumed to be spherical, of radii $R_1$ and $R_2$) then they will collide if and only if there is a $t$ such that $|x_1(t)-x_2(t)|<R_2+R_1$. (equation E)

If they are moving at constant speed $v_1$ and $v_2$, starting at initial positions $x_1^0$ and $x_2^0$ then this is achived iff $|x_1^0 - x2^0 + t (v_1-v_2)| < R_2 + R_1 $ has a solution for $t$ (it's a simple system of three inequalities).

Now if your objects are not spherical, the problem is more complicated (for example, they can spin). But if object one is contained within a sphere of radius $R_1$ (the same with object 2 and $R_2$) you can say that if (E) has no solution then they won't collide, and if your objects contain spheres of radii $r_1$ and $r_2$, then if (E) with $r_1$ and $r_2$ has a solution, they will collide.

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Calculating the shape of these objects is a question unto itself. Spheres are a good approximation for now though. –  Garzahd Apr 3 '12 at 15:54
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