[JohnASmith (OP)]

  As I understand it speed is relative therefore if two objects are moving towards each other their speed, relative to each other, is determined by how fast they approach each other.  A bus doing 30 MPH collides head-on with another bus doing 30 MPH the relative speed is 60 MPH.  The Hadron Collider sends atoms in both directions in a circular tunnel.  Those atoms pass each other repeatedly while they are accelerated to near the speed of light.  My questions are, how fast are those atoms traveling relative to each other and what is the relative speed just prior to collision?  Have they essentially passed the speed of light?

[PhysBang]

Adding speeds is not actually done as you suggest in the bus scenario. The relative speed (in a frame of reference not moving relative to one object) is always slightly less than the sum of the speeds relative to a third party.

[graham.d]

John, PhysBang is correct. It may seem counter-intuitive but it is a consequence of Special Relativity. You can observe object-A coming from your left at 0.9 times the speed of light (0.9c) and object-B coming from your right at 0.9c. From the perspective of someone on Object-B you would be coming towards him at 0.9c (as you may expect) but he would see Object-A coming towards him at 1.8c/(1+.9^2) = 1.8c/1.81 = 0.994475c (approx). No matter how fast the two objects are approaching each other from your perspective, their closing speed from the view of someone on one of the objects will always be less than the speed of light.

If you want to read more you could start here:

http://en.wikipedia.org/wiki/Velocity-addition_formula

but you need to understand Special Relativity to see how the result comes about.