University of Bahrain
PHYCS 222 Modern Physics
Department of Physics Dr. M. El-Hilo
First Semester 2011/2012
Ch. 2: Special theory of relativity
Task 1: Graph the variation of (v)=1/(1-(v/c)2)1/2 vs. v/c, what do you understand
from the variation of  with speed?.
Task 2: Graph the Galilean (non-relativistic) and the Lorentz (relativistic)
velocities, ux, relative to a fixed system for an object moving with ux=0.5c. What
can you conclude from these variations?
Task 3: Graph the decay of Muon with travelled distance; consider the followings:
N0=1000, Muon’s speed=0.98c, half-life as seen by the Muon is 1.52s, half-life
time as seen by us is 6.8s. Sketch the distance travelled by the Muon as seen by
the Muon and as seen by us. What are the proper time and length in this problem.
Why?.
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Task 4 Relativistic Dynamics: Velocities and Mass
Consider the following collision that is happing in the fixed frame: A particle of
mass m is moving along the positive x-axis at constant speed u1x=0.8c. Another
particle of the same mass is moving along the positive y-axis at a constant speed
u1y=0.6c. After they collide they become one mass of 2m and move at a speed u1f.
Using the non-relativistic momentum conservation calculations you will find that
u1f=0.5c since u1xf=0.4c and u1yf=0.3c. So in the K system, the collision appears as
follow
0.8c
y
0.3c
0.5c
y
0.6c
x
0.4c
x
Before Collision
After Collision
Collision as seen in the Lab, Non relativistic calculations
Before Collision
After collision
u1xi
u1yi
u1
u2xi
u2yi
u2
uxf
uyf
uf
0.8c
0
0.8c
0
0.6c
0.6c
0.4c
0.3c
0.5c
px=2m*0.4c-m*0.8c=0!!
py=2m*0.3c-m*0.6c=0!!
Now, a space ship is moving along the positive x-axis at v=0.866c and the
astronaut witness the same collision. Use relativistic velocities and describe how
the collision look like as seen by the astronaut, i.e what are the velocities before
and after the collision as seen by him (Sketch the collision as seen by the
astronaut). After you find u1xi, u1yi, u1i, u2xi, u2yi, u2i, uxf, uyi, and uf then ask the
astronaut to check the conservation of momentum, i.e. let him calculate px and
py, what did he find, which postulate does your answer violates?
List all your results for the velocities and speed before and after collision as seen
by the astronaut in a table similar to the above table.
u1xi
Collision as seen by someone moving to the right at v=0.866c
Before Collision
After collision
u1yi
u1
u2xi
u2yi
u2
uxf
uyf
uf
p1xi
p1yi
pxi=
p2xi
pyi=
p2yi
pxf
pxf=
pyf
pf
pyf=
2
Repeat all you previous calculations, in both the fixed and moving systems using
both relativistic velocities and mass. Do you get in both frames that the momentum
is conserved?
Collision as seen in the Lab, relativistic calculations
Before Collision
After collision
u1xi
u1yi
u1
u2xi
u2yi
u2
uxf
uyf
p1xi
p1yi
pxi=
p2xi
p2yi
pyi=
pxf
uf
pyf
pxf=
pyf=
Collision as seen by someone moving to the right at v=0.866c
Before Collision
After collision
u1xi
u1yi
u1
u2xi
u2yi
u2
uxf
uyf
p1xi
pxi=
p1yi
p2xi
pyi=
p2yi
pxf
pxf=
uf
pyf
pyf=
Task 5: Relativistic Energy
Plot the variation of the kinetic energy (K/E0) versus u/c for both the nonrelativistic and relativistic cases. What do you conclude from this?
Determine at what value of u/c the percentage error in using the non-relativistic
equation for the kinetic energy is equal 1%.
An electron is accelerated using a potential difference of 1000000V, what will be
the error in you calculations for the final speed if you use the non-relativistic
equation for the kinetic energy?
You did the e/m experiment and a potential difference of VA= 5000 V was used
to accelerate the electrons. Are you okay in this experiment with the nonrelativistic equation for the kinetic energy?
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Task 6: Task 6: Relativistic energy and momentum
An electron, photon and proton each has a momentum of 4GeV/c. For each
particle find its total energy, kinetic energy and speed. List your results in the
following table. What can you conclude from your calculations when Pc>E0?
Particle
Photon
Electron
Proton
In a system that is fixed relative to the three particles
P(GeV/c)
E (GeV)
KE (GeV)
u/c
4
4
4
The three particles are studied by someone in a moving system. In what moving
system, the proton will appear to have a momentum of 1GeV/c. In that frame of
reference, find the particle’s momentum, total energy, kinetic energy and speed
relative to the moving system. List your results in the following table.
In a system that is moving relative to the three particles (v=
Particle
P(GeV/c)
E (GeV)
KE (GeV) u/c
Photon
Electron
Proton
1
)
What can you conclude from all these calculations? What postulates that are
satisfied in these calculations?
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Task 7: All Relativistic
In a laboratory frame S, a proton of kinetic energy equals twice its rest energy
(i.e. Kp=2Eop) is moving along the positive x-axis collided with another proton
initially at rest. If we assume that after the collision, the incoming proton is
stopped and the resting proton move along the +ve x-axis. This collision is also
studied by an Alien moving along the +ve x-axis at speed v= 0.866c relative to
the laboratory frame.
V=0.866c
y
Lab
Incoming
proton
x
Resting
proton
Stopped ejected
proton proton
After
collision
Before
collision
Calculate the followings and list them in the following table:
Relative to the Alien
Relative to the laboratory frame (S)
Total
Energy
(GeV)
Kinetic
Energy
(GeV)
Momentu
m
(GeV/c)
Velocity
Total
Energy
(GeV)
Kinetic
Energy
(GeV)
Momentu
m
(GeV/c)
Velocity
Incident
proton
Resting
proton
E , P
before
collision
Stopped
proton
Ejected
proton
E , P
after
collision
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