Elements of Modern Physics
28
PROBLEMS
1. Show the two successive parallel Lorentz transformations are equivalent
to a single Lorentz transformation.
2. A rod of length l 0 in its rest frame, moves with velocity v parallel to itself.
Obtain the Lorentz contraction of the rod by calculating the time taken by
the rod to pass a point (use time dilation) and then multiplying this
time by v.
3. Obtain an expression for time dilation considering a clock with light bouncing
back and forth along the direction of relative velocity, and using the concept
of Lorentz contraction.
4. What is the visually observed rate of a clock which moves with velocity v
along the line of vision?
5. The incoming primary cosmic rays (mostly protons) create µ-mesons in
the upper atmosphere. The lifetime of µ-mesons at rest is 2.15 × 10
-6
s. If
the mean speed of the meson is 0.998 c, what fraction of the µ-mesons
created at a height of 20 km reach the sea level? What is the mean distance
travelled by the mesons before they decay?
6. A rod AB parallel to the x-axis, moves along the y-axis with velocity u.
Show that in a frame F′ which moves with velocity v along the x-direction,
this rod is inclined to the x′-axis at an angle
1
1/ 2
2
2
tan
.
2 1
−
−
uv
v
c
c
7. A ρ-meson of mass 760 MeV/c
2
decays at rest into two π-mesons of
mass 140 MeV/c
2
each. What is the relative velocity of the π-mesons
with respect to each other?
8. Show that when force f is not parallel to velocity u, the acceleration is in
general not parallel to either the force or the velocity.
9. A particle of mass M decays at rest into a particle of mass m and a
photon. What is the energy of the photon emitted? Apply this to
(a) Σ
+
(1189.4 MeV) → p (938.3 MeV) + γ, (b) H (2p) → H(1s) + γ, the
binding energy being 10.2 eV and 13.6 eV respectively.
10. A charged particle emits radiation when subjected to an external field.
This is known as bremsstrahlung. Show that energy-momentum
conservation does not allow a particle in isolation (no external forces) to
radiate. The argument is very simple in the centre of mass frame.
28
PROBLEMS
1. Show the two successive parallel Lorentz transformations are equivalent
to a single Lorentz transformation.
2. A rod of length l 0 in its rest frame, moves with velocity v parallel to itself.
Obtain the Lorentz contraction of the rod by calculating the time taken by
the rod to pass a point (use time dilation) and then multiplying this
time by v.
3. Obtain an expression for time dilation considering a clock with light bouncing
back and forth along the direction of relative velocity, and using the concept
of Lorentz contraction.
4. What is the visually observed rate of a clock which moves with velocity v
along the line of vision?
5. The incoming primary cosmic rays (mostly protons) create µ-mesons in
the upper atmosphere. The lifetime of µ-mesons at rest is 2.15 × 10
-6
s. If
the mean speed of the meson is 0.998 c, what fraction of the µ-mesons
created at a height of 20 km reach the sea level? What is the mean distance
travelled by the mesons before they decay?
6. A rod AB parallel to the x-axis, moves along the y-axis with velocity u.
Show that in a frame F′ which moves with velocity v along the x-direction,
this rod is inclined to the x′-axis at an angle
1
1/ 2
2
2
tan
.
2 1
−
−
uv
v
c
c
7. A ρ-meson of mass 760 MeV/c
2
decays at rest into two π-mesons of
mass 140 MeV/c
2
each. What is the relative velocity of the π-mesons
with respect to each other?
8. Show that when force f is not parallel to velocity u, the acceleration is in
general not parallel to either the force or the velocity.
9. A particle of mass M decays at rest into a particle of mass m and a
photon. What is the energy of the photon emitted? Apply this to
(a) Σ
+
(1189.4 MeV) → p (938.3 MeV) + γ, (b) H (2p) → H(1s) + γ, the
binding energy being 10.2 eV and 13.6 eV respectively.
10. A charged particle emits radiation when subjected to an external field.
This is known as bremsstrahlung. Show that energy-momentum
conservation does not allow a particle in isolation (no external forces) to
radiate. The argument is very simple in the centre of mass frame.