Faculty of Engineering and Physical Sciences
Department of Physics
Module PHY3038 – Special Relativity
Problems 2
1. An unstable nucleus, moving in a straight line, is seen to emit a particle. In the rest frame of
the nucleus the emitted particle moves with a velocity 67 c, at an angle of 60◦ with the direction
of motion of the nucleus. If a laboratory experimenter observes the particle to be emitted at an
angle of 30◦ to the original velocity of the nucleus, calculate the speed of the original nucleus.
2. Two frames S and S ′ are in relative motion along the x-axis with relative velocity v. An
observer O is located at the point x = a in the frame S and another observer O′ is located at
the point x′ = a in the frame S ′ . Show that the events
(i) the origin of S and S ′ passing each other, and
(ii) the observers in S and S ′ passing each other,
are separated in both frames by a time interval
(
1
a
1−
∆t = ∆t =
v
γv
)
′
,
but that the order in which the two events occur is reversed.
3. Two identical particles, of rest mass m0 , approach each other with equal and oppositely
directed velocities, u = βc. Show that, in the rest frame of one of the particles, the total energy
ε of the other particle is
ε=
(1 + β 2 )
m0 c2 .
(1 − β 2 )
4. A Kaon (K), travelling in a laboratory, dissociates into two pions (π) one pion being left at
rest. Express the energy of the original K and of the second pion in terms of the rest masses
mK and mπ of the particles involved.
5. A proton accellerator has an energy capability of 200 GeV. Assuming the proton rest energy
is 1 GeV, what is the most massive particle X which could be produced in the process
p + p −→ p + p + X
in which the energetic proton strikes a stationary target proton.
6. What is the minimum laboratory energy required of a proton beam, which strikes a stationary
target proton, to produce antiprotons (p̄) by the process
p + p −→ p + p + p + p̄ .
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You should assume the rest energies mp c2 = mp̄ c2 =1 GeV.
7. A beam of pions (rest mass mπ ) move in the x direction with a velocity u = 54 c. In the decay
of one such pion, a muon (rest mass mµ = 23 mπ ) and a neutrino are produced, the neutrino
( )
moving along the y–axis. Show that the muon velocity will make an angle tan−1 18 with the
x–axis and find the energy carried away by the neutrino.
8. A monochromatic wave, propagating in direction ⃗k in the frame S is written
f (⃗r, t) = A sin(⃗k · ⃗r − ωt) .
Using the transformation equations for x, y, z and t, in the case that two frames are in relative
motion along the x–axis with velocity v, show that, expressed in terms of the position and time
coordinates of the S ′ frame
f (⃗r ′ , t′ ) = A sin(⃗k ′ · ⃗r ′ − ω ′ t′ )
where
ω ′ = γ(ω − kx v)
kx′ = γ(kx − ωv/c2 )
ky′ = ky
kz′ = kz .
[That is, the four quantities k µ = (k 0 , k 1 , k 2 , k 3 ) = (ω/c, kx , ky , kz ) transform by the same
relationships as do the xµ = (ct, x, y, z).]
9. A particle of rest mass M0 is initially at rest in the laboratory. A second, lighter particle
of rest mass m0 , travelling with speed u collides with and combines with the larger mass.
Calculate the rest mass and the velocity of the massive particle produced in the collision, in
the laboratory frame.
10. A particle of rest mass m0 is initially at rest in the laboratory. A high energy photon,
with energy Eγ = m0 c2 (i.e. equal to the rest energy of the particle) is incident toward the
stationary particle. Calculate the energy of this system and the speed of the particle in the
centre of mass frame of the photon and particle.
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