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I'd like to know how to detect collisions exactly (the setting is in 2d). Specifically, given two polygonal rigid bodies with constant velocity and constant angular velocity how do I detect if they will collide. If so, I need the time of collision and moreover I need to compute the set of contact points.

I have yet to complete a physics engine (2d or 3d), so although I have some thoughts, I'm not going to comment on that particular area. However, I thought I'd mention that collision detection should be considered as a separate problem. True, a physics engine won't get far without a collision detection system, but a good collision detection library will be an independent module that can be used in any number of contexts (a physics engine being one of them).

So it sounds to me as if you're asking not only about 2d physics, but about 2d collision detection as well (correct me if I'm wrong). If so, you might specify what sort of objects you want to support (concave, convex, boxes, circles, segments, etc.), and whether you really need to incorporate angular as well as linear velocity (as this will increase the complexity immensely).