Equations (2-38) show that the four-momentum of a particle obeys a Lorentz transformation....

Equations (2-38) show that the four-momentum of a particle obeys a Lorentz transformation. If we sum momentum and energy over all particles in a system, we see that the tolal momentum and energy also constitute a four-vector. It follows that  is the same quantity in any frame of reference. Depending on what is known, this can be a quicker route to solving problems than writing out momentum and energy conservation equations. In the laboratory frame, a particle of mass m and energy Ei collides with another particle of mass in initially stationary, forming a single object of mass M. (a) Determine the frame of reference where the after-collision situation is as simple as possible, then determine the invariant in that frame, (b) Calculate the invariant before the collision in the laboratory frame in terms of Ei and m. (You will need to use  for the initially moving particle to eliminate its momentum.) Obtain an expression for M in terms of m and E-t. (c) Write out momentum and energy conservation equations in the laboratory frame, using Ei for the speed of the initially moving particle and uf for the speed of the final combined particle. Show that they give the same result for M in terms of m and Ei. (Note: The identity   will be veiy handy.)