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Simultaneous Events in Special Relativity
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Simultaneous Events in Special Relativity

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Questions and Answers

If a meson at rest decays 2 μs after it is created, what is its lifetime according to laboratory clocks when moving at 0.99c?

  • The same
  • 14 μs (correct)
  • 4.6 μs
  • 0.28 μs

    • What distance is required for the intensity of a beam of pi mesons traveling at speed v = βc to be halved if they have a half-life of T?

  • $(1 - \beta^2)^{1/2}vT$
  • $\beta vT$
  • $c\beta T[(1 + \beta)/(1 - \beta)]^{1/2}$
  • $c\beta T(1 - \beta^2)^{-1/2}$ (correct)

    • If a meson moving through a laboratory of length x at speed v decays after a lifetime T, what would its lifetime be if it were at rest in the laboratory?

  • $T(1 - v^2/c^2)^{-1/2}$
  • $T(1 - v/c)$
  • $T(1 - v/c)^{-1}$
  • $T(1 - v^2/c^2)^{1/2}$ (correct)

    • A meter stick moves sideways at 0.95c. What is its length according to measurements taken in the laboratory?

    <p>1.0 m</p> Signup and view all the answers

    At rest, pi mesons have a half-life of T. If a beam of pi mesons travels at speed v = βc, what happens to the intensity of the beam?

    <p>$T(1 - \beta^2)^{-1}$</p> Signup and view all the answers

    According to the theory of relativity, what is mass considered as?

    <p>A form of energy</p> Signup and view all the answers

    What happens to the mass of moving particles according to the theory of relativity?

    <p>They lose mass</p> Signup and view all the answers

    In high-speed collisions according to the theory of relativity, what is conserved?

    <p>Energy</p> Signup and view all the answers

    How is the length of a rod moving rapidly sideways (perpendicular to its length) affected?

    <p>It shortens along its length</p> Signup and view all the answers

    If a source emits light of frequency $f_0$ while moving to the right with speed $c/4$ relative to frame S, and a detector measures the frequency to be $f$, what can be concluded about the detector's movement?

    <p>The detector is moving to the right with a speed that is greater than $c/4$ relative to S</p> Signup and view all the answers

    If the kinetic energy of a free particle is much less than its rest energy, how is its kinetic energy related to the magnitude of its momentum?

    <p>Proportional to the square of the momentum's magnitude</p> Signup and view all the answers

    An electron with a speed of 0.95c has a momentum magnitude closest to which value?

    <p>8.3 × 10^-22 kg⋅m/s</p> Signup and view all the answers

    When light from a stationary spaceship is observed and then the spaceship moves directly away at high speed, what happens to the light seen by the observer?

    <p>A lower frequency and a longer wavelength than before</p> Signup and view all the answers

    In the scenario of a fast-moving train (v = 0.6c) with an engineer (E) at the front, a guard (G) at the rear, and a passenger (S') exactly halfway between them, what color signaling lamps do both E and G have?

    <p>Yellow signaling lamps</p> Signup and view all the answers

    When a source emits light of frequency $f_0$ and moves to the right with speed $c/4$ relative to frame S, what happens if a detector measures the frequency to be $f = f_0$?

    <p>The detector is not moving relative to S</p> Signup and view all the answers

    In an observation where light from a spaceship is initially seen and then it moves directly away at high speed, what phenomenon accounts for the change in light frequency?

    <p>Doppler effect</p> Signup and view all the answers

    If light from a stationary spaceship is initially observed and then the spaceship moves directly towards an observer at high speed, what will happen to the light seen by the observer?

    <p>A higher frequency and a shorter wavelength than before</p> Signup and view all the answers

    In reference frame S', what is the coordinate where the leading edges of two light flashes meet?

    <p>300 m</p> Signup and view all the answers

    In reference frame S', what is the time when the leading edges of two light flashes meet?

    <p>0.16 μs</p> Signup and view all the answers

    What is the speed of the clock that measures the proper time between the sending and receiving of the capsule relative to us?

    <p>0.95c</p> Signup and view all the answers

    In which direction does the clock that measures the proper time between the sending and receiving of the capsule travel?

    <p>Same direction as the spaceships</p> Signup and view all the answers

    What is the speed of frame S' with respect to frame S?

    <p>0.6c</p> Signup and view all the answers

    What is the speed of a particle that moves in the positive x direction as measured by an observer in S'?

    <p>0.4c</p> Signup and view all the answers

    According to an observer moving in the positive x direction, how do two simultaneous events occurring on the y axis of a reference frame S appear?

    <p>The events are simultaneous</p> Signup and view all the answers

    A train traveling at 0.6c passes a station. According to both the station master (S) and an observer (S') on the train, which of the following statements is true about the signals sent by the engineer (E) and the guard (G)?

    <p>E and G sent their signals simultaneously from different distances</p> Signup and view all the answers

    According to the observer (S') on the train, how do the signals sent by the engineer (E) and the guard (G) appear when the train is passing the station?

    <p>The signals arrive simultaneously, with G sending his signal first</p> Signup and view all the answers

    A station master (S) observes a train traveling at 0.6c. According to S, how do the signals sent by the engineer (E) and the guard (G) appear?

    <p>The signals arrive simultaneously, with E sending his signal first</p> Signup and view all the answers

    A train traveling at 0.6c passes a station. According to the observer (S') exactly halfway between the engineer (E) and the guard (G), how do the signals sent by E and G appear?

    <p>The signals arrive simultaneously, with E sending his signal first</p> Signup and view all the answers

    A train traveling at 0.6c passes a station. According to the station master (S), how do the distances from the engineer (E) and the guard (G) to the station compare when they send their signals?

    <p>G is further away from the station than E</p> Signup and view all the answers

    Study Notes

    Relativity

    • Meson decay:
      • At rest, decays 2 μs after creation
      • When moving at 0.99c, its lifetime is 14 μs according to laboratory clocks
    • Pi mesons:
      • Half-life is T at rest
      • When traveling at speed v = βc, the distance in which the beam intensity is halved is cβT(1 – β^2)^(-1/2)
    • Time dilation:
      • A meson moving at speed v through a laboratory of length x decays after a lifetime T
      • If the meson were at rest in the laboratory, its lifetime would be T(1 – v^2/c^2)^1/2
    • Length contraction:
      • A meter stick moving sideways at 0.95c has a length of 0.31 m according to laboratory measurements
    • Relativity principles:
      • Mass is a form of energy
      • Moving particles do not lose mass
      • Momentum is conserved in high-speed collisions
      • A rod moving rapidly sideways is shorter along its length
    • Kinetic energy:
      • If the kinetic energy of a free particle is much less than its rest energy, its kinetic energy is proportional to the square of the magnitude of its momentum
    • Particle momentum:
      • An electron with a speed of 0.95c has a momentum of 8.3 × 10^(-22) kg â‹… m/s
      • A particle with mass m and momentum 2mc has a speed of 0.89c
    • Photon energy:
      • A particle with zero mass and energy E carries momentum E/c
    • Doppler shift:
      • When a source moves to the right with speed c/4 relative to a detector, the frequency measured by the detector is greater than the proper frequency
      • When a spaceship moves away from an observer at high speed, the light seen by the observer has a lower frequency and a longer wavelength
    • Special relativity:
      • A clock that measures the proper time between the sending and receiving of a message capsule travels at 0.95c relative to the observer
      • Two events occur simultaneously at separated points on the y-axis of a reference frame; according to an observer moving in the positive x direction, the events are simultaneous

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    Test your understanding of events occurring simultaneously in special relativity from the perspective of an observer moving in the positive x direction. Determine the correct sequence of events and the impact of observer's speed on simultaneity.

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