#3 Given time shift of aircraft, calculate speed.
Time difference = 4.00E-09 seconds, c= 3.00E+08
Distance 3.00E+06 meters in earth frame
a. Earth time interval is longer, stationary observer.
b. Pilot sees 3000 km as length contracted.
Both observers measure same value for v
t(earth) = 3000/v
t(pilot) = (3000)sqrt(1-v^2/c^2)/v
3000/v(1 - sqrt(1-v^2/c^2)) = 4ns
Use first two terms of binomial expansion:
sqrt(1-a) approx= 1 - a/2
3000000(v/2c^2) = 4 ns
v = 239.7 m/s
6. Proper lifetime of particle
Proper lifetime means lifetime in rest frame of particle
v =.99c with respect to lab frame
Lab frame distance = 0.001 m
Particle frame distance = 0.000141
Two frames agree on the 0.99c velocity
Particle frame time = distance/velocity = 4.75 E-13 sec
8. Muon and pion
a. Muon proper lifetime = 2.20E-06 seconds
speed = .99c
Distance before decay = 652.96 meters in muon frame
The relativity factor is = 7.0888
In the lab frame, this would be 4629 meters in lab frame.
b. Charged pion lifetime = 2.60E-08 seconds
speed = .99c
relativity factor = 7.0888
Lab frame distance = 54.7 meters
9. Twin paradox
a. Earthbound twin determines time to be shorter in travelers frame.
b. In the absence of acceleration, the traveler would measure the earthbound twin's time to be shorter.
c. The acceleration is in the travelers frame, so the earth frame is considered to be the inertial frame.
The traveling twin would be younger.
13. Momentum and speed of a 1 GeV proton.
From the relativistic kinetic energy expression, the speed must be 0.875 c From the relativistic momentum expression, its momentum is then = 9.06E-19 kg m/s
At this speed, the non-relativistic momentum expression would give 4.38E-19 kg m/s
which would be in error by 51%
Another approach is to use pc =sqrt(E^2 - (m0c^2)^2) and express p in MeV/c
Must note that the E in this expression includes both KE and the rest mass energy.
pc=sqrt((1GeV +.938GeV)^2-(.938 GeV)^2)) = 1.696 GeV
The momentum can then be expressed as p=1.70 GeV/c
14. Radius of curvature of 1 MeV electron in 1 Tesla magnetic
Force equation mv^2/r = evB, so r = mv/eB, but relativistic momentum must be used
From the relativistic kinetic energy expression, the velocity must be= 0.941 c
for the kinetic energy to be 1 MeV.
From the relativistic momentum relationship, p = 7.59E-22 kg m/s
e= 1.60E-19 C
Resulting radius = 4.74E-03 m
17. Calculated mass of alpha particle
mass of proton= 938.3 MeV in energy equivalents
mass of neutron= 939.6 MeV
Binding energy = 28.4 MeV
mass of alpha= 3727.4 MeV
mass of alpha= 6.64E-27 kg
18. Carbon-14 decay energy
From Appendix K of Rohlf
mass of carbon 13040.976 MeV
mass of nitrogen 13040.309 MeV
mass of electron 0.511 MeV
energy release 0.156 MeV
21. Electron KE=mass energy, find photon energy for photon with same
Electron mass energy = 0.511 MeV
KE of electron = 0.511 MeV requires speed of 0.866 c
The momentum of this electron is then = 4.73E-22 kg m/s
The momentum of a photon = E/c, so E=pc = 1.42E-13 joules
22. Speed to red-shift to 1% of frequency
To shift frequency to 1% of its source frequency requires a recession
velocity of 0.9998 c
29. Z-zero decay into electron and positron
Mass energy of Z-zero = 91.2 GeV
Particle is at rest, so electron and positron must have equal but
oppositely directed momenta.
Energy of each is 45.6 GeV p=45.6 GeV/c
For this extreme relativistic case, the mass energy can be neglected
in calculating the momentum, and the photon expression p=E/c used.
With this approximation, p= 152.1014009 eV s/m
The required speed is so close to c that you can't take the square root
to calculate c.
The relativistic gamma factor is = 45,600MeV/.511MeV =
gamma = 89236.8
(use b and g for beta and gamma since Greek letters can't be depended upon.)
We can assess the departure from c with the fractional departure from c, 1 -v/c
The trick is to find an expression for 1-v/c = 1-b which we can evaluate.
Since g^2 = 1/(1-b^2) = 1/((1-b)(1+b)), then we can write
1-b = 1/((1+b)g^2) or approximately: 1-b = 1/2g^2
This gives 1-b= 6.28E-11
36. Back-scattered 10GeV photon
a) High energy collisions like this need to be viewed in the center-of-mass
frame. The photon energy does not change in the collision as viewed in the
COM frame. Transforming back to the laboratory frame, the electron has
essentially all the energy. Take electron energy = 10 GeV, and the photon
energy can be approximated by:
E' = mc^2 E/(mc^2 + 2E) approx= mc^2/2 = .511 MeV/2 = .26 MeV
b) Again the collision is examined in the COM frame. This time
the photon has
essentially all the energy. The photon energy is about 10 GeV; the electron
energy is about mc^2
Relativistic projectile problem.
Enterprise traveling at 0.8c as seen by external observer
Projectile fired at 0.6c as seen by Enterprise
Klingon vessel approaches at 0.7c in opposite direction as seen by external observer
Projectile speed as seen by external observer = 0.9459
Approach speed of Klingons as seen by Enterprise = -0.9615 c
Projectile speed as seen from Klingon ship = 0.9902 c