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Special Relativity

Time Dilation

t = \frac{t_0}{\sqrt{1-\frac{v^2}{c^2}}}

Length Contraction

L = L_0 \sqrt{1-\frac{v^2}{c^2}}


Relativistic Mass

m = \frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}

Relativistic Momentum

P=mv=\frac{m_0 v}{\sqrt{1-\frac{v^2}{c^2}}}

Relativistic Newton's Second Law

F = \frac{d}{dt}(\frac{m_0 v}{\sqrt{1-\frac{v^2}{c^2}}})

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

F_R = -bv
\gamma = \frac{b}{m}

Solution 1

mv_x = -bv_x
mv_y = -mg-bv_y
t = \frac{2v_{0y}}{g}( 1 + \frac{\gamma v_{0y}}{g} )

Mass and Energy

Total Energy
mc^2 = m_0 c^2 + k
(k = kinetic energy) Rest Energy
E = m_0 c^2
Total Energy
E = m_ c^2 = \frac{m_0 c^2}{\sqrt{1-\frac{v^2}{c^2}}}
E^2 = m^2_0 c^4 + p^2 c^2

Problem 2

An electron has total energy equal to four times its rest energy. What is the momentum of the electron in units of mec?

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

v = \frac{4}{5}c

The lifetime of a muon in its rest frame is

\tau = 2.2 \times 10^{-6} \mathrm{s}

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. T1 = 3T2). The ratio m1/m2 of the masses of the two planets is:

Solution 7

Problem 8

A distant galaxy is observed to have its hydrogen-β 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:

\frac{580}{434} \approx \frac{4}{3}

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 x-axis, 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 x-axis separated in time by Δ t and in space by Δ x. In another inertial reference frame S', moving in the x-direction relative to S, the 2 events could occur at the same time under which if any, of the following conditions?

  1. For any values of Δ x and Δ t
  2. Only if | Δ x / Δ t | < c
  3. Only if | Δ x / Δ t | > c
  4. Only if | Δ x / Δ t | = c
  5. 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

  1. 2.8E8 m/s toward Earth
  2. 2.8E8 m/s away from Earth
  3. 2.4E8 m/s toward Earth
  4. 2.4E8 m/s away from Earth
  5. 12E8 m/s away from Earth

Solution 13


Problem 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/c2)?

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 mc2?

Solution 16

Problem 17

A positive kaon (K+) has a rest mass of 494 MeV/c2 whereas a proton has a rest mass of 938 MeV/c2. 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 c2 referes to the rest mass. So the rest mass of the kaon and proton are

E_{Krest} = m_K c^2
E_{Prest} = m_P c^2

and the total mass mass of the kaon and proton are

E_{Ktot} = \gamma m_K c^2
E_{Ptot} = \gamma m_P c^2

In this problem, the total energy of the kaon is equal to the rest energy of the proton

E_{Ktot} = E_{Prest}
\gamma m_k c^2 = m_p c^2
The math becomes
Solve for v

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 space-time 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?

  1. X does not spin
  2. The spin of X equals the spin of a photon
  3. The rest mass of X is zero
  4. The charge of X is carried on its surface
  5. X cannot be detected

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 π0 meson (rest-mass energy 135 MeV) is moving with velocity 0.8 c k̂ in the laboratory rest frame when it decays into two photons γ1 and γ2. In the π0 rest frame, γ1 is emitted forward and γ2 is emitted backward relative to the π0 direction of flight. What is the velocity of γ2 in the laboratory rest frame?

Solution 23

Problem 24

Tau leptons are observed to have an average half-life of Δ t1 in the frame S1 in which the leptons are at rest. In an inertial frame S2, which is moving at a speed v12 relative to S1, the leptons are observed to have an average half-life of Δ t2. in another inertial reference frame S3, which is moving at a speed v13 relative to S1 and v23 relative to S2, the leptons have an observed half-life of Δ t3. Which of the following is a correct relationship among two of the half lives, Δ t1, Δ t2, and Δ t3.
\Delta t_2=\Delta t_1\sqrt{1-v_{12}^2/c^2}
\Delta t_1=\Delta t_3\sqrt{1-v_{13}^2/c^2}
\Delta t_2=\Delta t_3\sqrt{1-v_{23}^2/c^2}
\Delta t_3=\Delta t_2\sqrt{1-v_{23}^2/c^2}
\Delta t_1=\Delta t_2\sqrt{1-v_{23}^2/c^2}

Solution 24


Problem 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

  1. m c
  2. 10 m c
  3. 70 m c
  4. 100 m c
  5. 104 m c

Solution 25

Problem 26

A free electron (rest mass me = 0.5 MeV/c2) 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.)
  1. x' = 4x
    y' = y
    z' = z
    t' = .25t
  2. x' = x - 0.75t
    y' = y
    z' = z
    t' = t
  3. x' = 1.25x - 0.75t
    y' = y
    z' = z
    t' = 1.25t - 0.75x
  4. x' = 1.25x - 0.75t
    y' = y
    z' = z
    t' = 0.75t - 1.25x
  5. None of the above

Solution 27

Lorentz transformations are
x' = \gamma (x-vt)
t' = \gamma (t - \frac{vx}{c^2})

Problem 28

A lump of clay whose rest mass is 4 kilograms is traveling at three-fifths 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 2.5E-8 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 xy-plane 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

E = \frac{\sigma}{2 \epsilon_0}

where the charge density

\sigma = \frac{Q}{Area} = \frac{Q}{X \cdot Y}
but with respect to the second observer, the length in X must be shortened. So find the shortened length by
X' = \sqrt{1-\frac{v}{c})^2} \cdot X
Then the new charge density is
\sigma = \frac{Q}{\sqrt{1-(v/c)^2} \cdot X \cdot Y}
Plug this into the Energy equation
E = \frac{Q}{\sqrt{1-(v/c)^2} \cdot X \cdot Y} \cdot \frac{1}{2 \epsilon_0}

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^2 = (cdt)^2 - (dx)^2

dS is invariant. You have

dS^2 = dS'^2
ct = 4 c minutes

Problem 34

Which of the following reasons explains why a photon cannot decay to an electron an a positron in free space?

\gamma \rightarrow e^{+} + e^{-}
  1. Charge and lepton number ar enot both conserved
  2. Linear momentum and angular momentum are not both conserved
  3. Linear momentum and energy are not both conserved
  4. Parity and strangeness are not both conserved
  5. Angular momentum and parity are not both conserved

Solution 34