Problems on Four-vectors
Problem 2.1:
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.
[Solution]
Problem 2.2:
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.
[Solution]
Problem 2.3:
Prove that if
A and
B are 4-vectors,
A.B their inner product is independent of the frame in
which it is calculated.
[Solution]
Problem 2.4:
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.
[Solution]
Problem 2.5:
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.
[Solution]
in the lab frame and a particle of mass M moving with velocity 
also
in the lab frame.
[Solution]
[Solution]
