# Special Relativity: Dynamics

### Problems on Four-vectors

Problem : Find the inner product of the 4-momenta for the following two particles: a particle of mass m moving with speed in the lab frame and a particle of mass M moving with velocity also in the lab frame.

Problem : Calculate the same inner product as in the previous question, but now in a frame moving with one of the particles (or, if you already did it in such a frame, calculate it in the lab frame). Check that the result is the same.

Problem : Prove that if A and B are 4-vectors, A.B their inner product is independent of the frame in which it is calculated.

Problem : Derive the velocity addition formula using the invariance of the 4-velocity inner product. In other words, if in frame A, B moves to the right with speed v , and C moves to the left with speed u , find w , the speed of B with respect to C.

Problem : Again using the invariance of the inner product, determine the speed of one particle as observed by the other as two particles approach each other with speed v along trajectories separated by an angle 2θ , as shown in the figure below.

Particle approaching each other at an angle.

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