Physics 585 - Problem Set 1
Relativistic Kinematics I
1) As the outlaws escape in their getaway car, which goes at (3/4)c, the cop fires a bullet from the pursuit car, which only goes at (1/2)c. The muzzle velocity (speed relative to the gun) of the bullet is (1/3)c. Does the bullet reach its target,
(a) according to prerelativistic (classical) physics ? (5)
(b) according to relativistic physics ? (5)
2) Show explicitly that the quantity I = (x0)2 (x1)2 (x2)2 (x3)2 is invariant under the Lorentz transformation. (10)
3) Prove that the invariant quantity hµhµ has the value c2. (10)
4) Consider an experiment in which particle a (with mass Ma) collides with particle b (mass Mb), producing particle c (mass Mc) and particle d (mass Md).
a + b à c + d
Assuming that 4-momentum is conserved in system S i.e.
Use the Lorentz transformation to show that 4-momentum is also conserved in S', a reference frame moving at constant velocity with respect to S. (15)
5) A pion (M = 140 MeV/c2) decays at rest into a muon (M = 106 MeV/c2) plus a neutrino (assumed to be zero mass). What is the speed of the muon ? (20)
6) Calculate the threshold energy for the creation of a single pion in the reaction below in the centre of mass frame (CM) of the initial particles. The threshold condition is that all the final state products are at rest in the CM frame.
p + p > p + p + p0
Repeat the calculation in the reference frame in which the target proton is initially at rest. This is usually called the LAB frame. (20)
7) A “charmed” meson D0 decays via D0 -> K-p+ , 3 mm from its production point. The total energy of the decay products is 30 GeV. Mass of the D0 = 1.864 GeV/c2. (Bettini, Ch. 1#18)
(a) What is the lifetime of the D0 in its rest frame ? (10)
(b) What is the momentum of the p+ in the D0 rest frame ? (10)