Special Relativity
Time Dilation
Length Contraction
Dynamics
Relativistic Mass
Relativistic Momentum
Relativistic Newton's Second Law
Problem 1
Determine the corrected value for the time of flight t of a projectile near the Earth's surface (in 2 dimensions) subject to a resistive force
Solution 1
Mass and Energy
Total EnergyProblem 2
An electron has total energy equal to four times its rest energy. What is the momentum of the electron in units of m_{e}c?
Solution 2
Problem 3
Two spaceships approach Earth with equal speeds, as measured by an observer on Earth, but from opposite directions. A meterstick on one spaceship is measured to be 60cm long by an occupant of the other spaceship. What is the speed of each spaceship (in units of c), as measured by the observer on Earth?
Solution 3
Problem 4
A meter stick with a speed of 0.8c moves past an observer. In the observer's reference frame, how many nanoseconds does it take the stick to pass the observer?
Solution 4
Problem 5
A beam of muons travels through the laboratory with speed
The lifetime of a muon in its rest frame is
What is the mean distance (in meters) traveled by the muons in the laboratory frame?
Solution 5
Problem 6
A particle of mass M decays from rest into 2 particles. 1 particle has mass m and the other particle is massless. What is the momentum of the massless particle?
Solution 6
Problem 7
Astronomers observe two separate solar systems, each consisting of a planet orbiting a sun. The 2 orbits are circular and have the same radius R. It is determined that the planets have angular momenta of the same magnitude L about the suns, and tha tthe orbital periods are in the ratio of three to one (i.e. T_{1} = 3T_{2}). The ratio m_{1}/m_{2} of the masses of the two planets is:
Solution 7
Problem 8
A distant galaxy is observed to have its hydrogenβ line shifted to a wavelength of 580 nm, away from the laboratory value of 434 nm. Which of the following gives the approximate velocity of recession of the distant galaxy? Note:
Solution 8
Problem 9
An observer O at rest midway between 2 sources of light at x = 0 and x = 10m observes the two sources to flash simultaneously. According to a second observer O' moving at a constant speed parallel to the xaxis, one source of light flashes 13ns before the other. What is the speed of O' relative to O? How many c?
Solution 9
Problem 10
If the total energy of a particle of mass m is equal to twoce its rest energy then what is the magnitude of the particles' relativistic momentum? How many mc?
Solution 10
Problem 11
If a charged pion that decays in 10^{8} second in its own rest frame is to travel 30 meters in the laboratory before decaying, the pion's speed must be most nearly how many m/s?
Solution 11
Problem 12
In an inertial reference frame S, 2 events occur on the xaxis separated in time by Δ t and in space by Δ x. In another inertial reference frame S', moving in the xdirection relative to S, the 2 events could occur at the same time under which if any, of the following conditions?
 For any values of Δ x and Δ t
 Only if  Δ x / Δ t  < c
 Only if  Δ x / Δ t  > c
 Only if  Δ x / Δ t  = c
 Under no condition
Solution 12
Problem 13
The ultraviolet Lyman alpha line of hydrogen with wavelength 121.5 nanometers is emitted by an astronomical object. An observer on Earth measures the wavelength of the light received from the object to be 607.5 nanometers. The obesrver can conclude that the object is moving with a radial velocity of
 2.4E8 m/s away from Earth
 2.8E8 m/s away from Earth
 2.8E8 m/s toward Earth
 2.4E8 m/s toward Earth
 12E8 m/s away from Earth
Solution 13
BProblem 14
A particle leaving a cyclotron has a total relativistic energy of 10GeV and a relativisitc momentum of 8 GeV/c. What is the rest mass of this particle in (GeV/c^{2})?
Solution 14
Problem 15
A tube of water is traveling at 1/2 c relative to the lab frame when a beam of light traveling in the same direction as the tube neters it. What is the speed of light in the water relative to the lab frame? (The index of refraction of water is 4/3)
Solution 15
Problem 16
A photon strikes an electron of mass m that is initially at rest, creating an electron positron pair. The photon is destroyed and the positron and 2 electrons move off at equal speeds along the initial direction of the photon. The energy of the photon was how many mc^{2}?
Solution 16
Problem 17
A positive kaon (K^{+}) has a rest mass of 494 MeV/c^{2} whereas a proton has a rest mass of 938 MeV/c^{2}. If a kaon has a total energy that is equal to the proton rest energy, the speed of the kaon is most nearly how many c?
Solution 17
Remember the m in E = m c^{2} referes to the rest mass. So the rest mass of the kaon and proton are
and the total mass mass of the kaon and proton are
In this problem, the total energy of the kaon is equal to the rest energy of the proton
Problem 18
Two observers O and O' observe 2 events, A and B. The observers have a constant relative speed of 0.8c. In unites such that the speed of light is 1, observer O obtained the following coordinates:
What is the length of the spacetime interval between these two events as measured by O'?
Solution 18
Problem 19
If a newly discovered particle X moves with a speed equal to the speed of light in a vacuum, then which of the following must be true?
 X does not spin
 X cannot be detected
 The rest mass of X is zero
 The spin of X equals the spin of a photon
 The charge of X is carried on its surface
Solution 19
Problem 20
A car of rest length 5 meters passes through a garage of rest length 4 meters. Due to the relativistic Lorentz contraction, the car is only 3 meters long in the garage's rest frame. There are doors on both eneds of the garage, which open automatically when the front of the car reaches them, and close automatically when the rear passes them. The opening or closing of each door requires a negligible amount of time.
What is the velocity of the car in the garage's rest frame?
Solution 20
Problem 21
Using the problem above, what is the length of the garage in the car's rest frame?
Solution 21
Problem 22
Using the problem above, was the car ever inside a closed garage?
Solution 22
Problem 23
A π^{0} meson (restmass energy 135 MeV) is moving with velocity 0.8 c k̂ in the laboratory rest frame when it decays into two photons γ_{1} and γ_{2}. In the π^{0} rest frame, γ_{1} is emitted forward and γ_{2} is emitted backward relative to the π^{0} direction of flight. What is the velocity of γ_{2} in the laboratory rest frame?
Solution 23
Problem 24
Tau leptons are observed to have an average halflife of Δ t_{1} in the frame S_{1} in which the leptons are at rest. In an inertial frame S_{2}, which is moving at a speed v_{12} relative to S_{1}, the leptons are observed to have an average halflife of Δ t_{2}. in another inertial reference frame S_{3}, which is moving at a speed v_{13} relative to S_{1} and v_{23} relative to S_{2}, the leptons have an observed halflife of Δ t_{3}. Which of the following is a correct relationship among two of the half lives, Δ t_{1}, Δ t_{2}, and Δ t_{3}.Solution 24
BProblem 25
A monoenergetic beam consists of unstable particles with total energies 100 times their rest energy. If the particles have rest mass m, their momentum is most nearly
 m c
 10 m c
 70 m c
 100 m c
 10^{4} m c
Solution 25
Problem 26
A free electron (rest mass m_{e} = 0.5 MeV/c^{2}) has a total energy of 1.5 MeV. What is the momentum p in units of MeV/c?
Solution 26
Problem 27
Which of the following is a Lorentz transformation? (Assume a system of units such that the velocity of light is 1.)
x' = 4x
y' = y
z' = z
t' = .25t

x' = x  0.75t
y' = y
z' = z
t' = t

x' = 1.25x  0.75t
y' = y
z' = z
t' = 1.25t  0.75x

x' = 1.25x  0.75t
y' = y
z' = z
t' = 0.75t  1.25x
 None of the above
Solution 27
Lorentz transformations areProblem 28
A lump of clay whose rest mass is 4 kilograms is traveling at threefifths the speed of light when it collides head on with an identical lump going the opposite direction at the same speed. If the 2 lumps stick together and no energy is radiated away, what is the mass of the composite lump?
Solution 28
Problem 29
An atom moving at speed 0.3c emits an electron along the same direction with speed 0.6c in the internal rest frame of the atom. The speed of the electron in the lab frame is equal to
Solution 29
Problem 30
What is the speed of a particle having a momentum of 5MeV/c and a total relativistic energy of 10 MeV?
Solution 30
Problem 31
The half life of a π^{+} meson at rest is 2.5E8 seconds. A beam of π^{+} is generated at a point 15m from a detector. Only half of the π^{+} mesons live to reach the detector. The speed of the π^{+} mesons is:
Solution 31
Problem 32
The infinite xyplane is a nonconducting surface with surface charge density σ as measuered by an observer at rest on the surface. What is the velocity of the second observer?
Solution 32
The electric field of 2 charged sheets can be determined by Gauss's Law
where the charge density
Problem 33
In inertial frame S, 2 events occur at the same instant in time and 3 c∙minutes apart in space. In inertial frame S', the same events occur at the 5 c∙minutes apart. What is the time interval (in minutes) between the events in S'?
Solution 33
The spacetime interval is defined by the metric that negates spatial and time variables as
dS is invariant. You have
Problem 34
Which of the following reasons explains why a photon cannot decay to an electron an a positron in free space?
 Linear momentum and angular momentum are not both conserved
 Angular momentum and parity are not both conserved
 Charge and lepton number ar enot both conserved
 Parity and strangeness are not both conserved
 Linear momentum and energy are not both conserved