Deck 37: Relativity

Question 1

Pi mesons at rest have a half-life of T. If a beam of pi mesons is traveling at a speed of v = $\beta$ c, the distance in which the intensity of the beam is halved is:

A) c $\beta$ T(1 - $\beta$ ²)^-1/2
B) c $\beta$ T[(1 + $\beta$ )/(1 - $\beta$ )]^1/2
C) $\beta$ vT
D) (1 - $\beta$ ²)^1/2vT
E) none of the above

Answer 1

c $\beta$ T(1 - $\beta$ ²)^-1/2

Question 2

A meson when at rest decays 2 $\mu$ s after it is created. If moving in the laboratory at 0.99c, its lifetime according to laboratory clocks would be:

A) the same
B) 0.28 s
C) 14 $\mu$ s
D) 4.6 s
E) none of these

Answer 2

14 $\mu$ s

Question 3

A clock is moving along the x axis at 0.6c. It reads zero as it passes the origin (x = 0). When it passes the x = 180 m mark on the x axis the clock reads:

A) 0.60 $\mu$ s
B) 0.80 $\mu$ s
C) 1.00 $\mu$ s
D) 1.25 $\mu$ s
E) 1.67 $\mu$ s

Answer 3

0.80 $\mu$ s

Question 4

A millionairess was told in 1992 that she had exactly 15 years to live. However, if she travels away from the Earth at 0.8 c and then returns at the same speed, the last New Year's day the doctors expect her to celebrate is:

A) 2001
B) 2003
C) 2007
D) 2010
E) 2017

Question 5

A measurement of the length of an object that is moving relative to the laboratory consists of noting the coordinates of the front and back:

A) at different times according to clocks at rest in the laboratory
B) at the same time according to clocks that move with the object
C) at the same time according to clocks at rest in the laboratory
D) at the same time according to clocks at rest with respect to the fixed stars
E) none of the above

Question 6

An observer notices that a moving clock runs slow by a factor of exactly 10. The speed of the clock is:

A) 0.100c
B) 0.0100c
C) 0.990c
D) 0.900c
E) 0.995c

Question 7

The spaceship U.S.S. Enterprise, traveling through the galaxy, sends out a smaller explorer craft that travels to a nearby planet and signals its findings back. The proper time for the trip to the planet is measured by clocks:

A) on board the Enterprise
B) on board the explorer craft
C) on Earth
D) at the center of the galaxy
E) none of the above

Question 8

Spaceship A, traveling past us at 0.7c, sends a message capsule to spaceship B, which is in front of A and is traveling in the same direction as A at 0.8c relative to us. The capsule travels at 0.95c relative to us. A clock that measures the proper time between the sending and receiving of the capsule travels:

A) in the same direction as the spaceships at 0.7c relative to us
B) in the opposite direction from the spaceships at 0.7c relative to us
C) in the same direction as the spaceships at 0.8c relative to us
D) in the same direction as the spaceships at 0.95c relative to us
E) in the opposite direction from the spaceships at 0.95c relative to us

Question 9

Two events occur simultaneously at separated points on the y axis of reference frame S. According to an observer moving in the positive x direction:

A) the event with the greater y coordinate occurs first
B) the event with the greater y coordinate occurs last
C) either event might occur first, depending on the observer's speed
D) the events are simultaneous
E) none of the above

Question 10

A meter stick moves sideways at 0.95c. According to measurements taken in the laboratory, its length is:

A) 0
B) 0.098 m
C) 0.31 m
D) 3.2 m
E) 1.0 m

Question 11

A basic postulate of Einstein's theory of relativity is:

A) moving clocks run more slowly than when they are at rest
B) moving rods are shorter than when they are at rest
C) light has both wave and particle properties
D) the laws of physics must be the same for observers moving with uniform velocity relative to each other
E) everything is relative

Question 12

The length of a meter stick moving at 0.95c in the direction of its length is measured by simultaneously marking its ends on a stationary axis. As measured by clocks moving with the stick, the time interval between the making of the back mark and the making of the front mark is:

A) 0
B) 3.1 * 10^-10 s
C) 1.0 * 10^-9 s
D) 3.2 * 10^-9 s
E) 1.0* 10^-8 s

Question 13

As we watch, a spaceship passes us in time $\Delta$ t. The crew of the spaceship measures the passage time and finds it to be $\Delta$ t'. Which of the following statements is true?

A) $\Delta$ t is the proper time for the passage and it is smaller than $\Delta$ t'
B) $\Delta$ t is the proper time for the passage and it is greater than $\Delta$ t'
C) $\Delta$ t' is the proper time for the passage and it is smaller than $\Delta$ t
D) $\Delta$ t' is the proper time for the passage and it is greater than $\Delta$ t
E) None of the above statements are true.

Question 14

A meter stick moves in the direction of its length through a laboratory. According to measurements taken in the laboratory, its length is 0.31 m. The speed of the meter stick relative to the laboratory is:

A) 0.096c
B) 0.31c
C) 0.69c
D) 0.83c
E) 0.95c

Question 15

A train traveling very fast (v = 0.6c) has an engineer (E) at the front, a guard (G) at the rear and an observer (S') exactly half way between them. Both E and G are equipped with yellow signaling lamps. The train passes a station, closely observed by the station master (S). Both E and G use their lamps to send signals. According to both S and S' these signals arrive simultaneously at the instant S' is passing S. According to S':

A) E and G sent their signals simultaneously from different distances
B) G sent his signal before E and from further away
C) G sent his signal before E but was the same distance away
D) E sent his signal before G and from further away
E) none of the above

Question 16

A certain automobile is 6 m long if at rest. If it is measured to be 4/5 as long, its speed is:

A) 0.1c
B) 0.3c
C) 0.6c
D) 0.8c
E) > 0.95c

Question 17

A rocket ship of rest length 100 m is moving at speed 0.8c past a timing device which records the time interval between the passage of the front and back ends of the ship. This time interval is:

A) 0.20 $\mu$ s
B) 0.25 $\mu$ s
C) 0.33 $\mu$ s
D) 0.52 $\mu$ s
E) 0.69 $\mu$ s

Question 18

Two events occur simultaneously on the x axis of reference frame S,one at x = -a and the other at x = +a. According to an observer moving in the positive x direction:

A) the event at x = +a occurs first
B) the event x = -a occurs first
C) either event might occur first, depending on the value of a and the observer's speed
D) the events are simultaneous
E) none of the above

Question 19

A meson moving through a laboratory of length x at a speed v decays after a lifetime T as measured by an observer at rest in the laboratory. If the meson were at rest in the laboratory its lifetime would be:

A) T(1 - v/c)
B) T(1 - v/c)^-1
C) T(1 - v²/c²)^-1/2
D) T(1 - v²/c²)^1/2
E) (T - vx/c²)(1 - v²/c²)^-1/2

Question 20

The proper time between two events is measured by clocks at rest in a reference frame in which the two events:

A) occur at the same time
B) occur at the same coordinates
C) are separated by the distance a light signal can travel during the time interval
D) occur in Boston
E) satisfy none of the above

Question 21

Two independent events occur 100 m apart with an intervening time interval of 0.42 $\mu$ s. The proper time in $\mu$ s between the events is:

A) 0
B) 0.16
C) 0.26
D) 0.42
E) 0.69

Question 22

A consequence of Einstein's theory of relativity is:

A) moving clocks run faster than when they are at rest
B) moving rods are shorter than when they are at rest
C) light has both wave and particle properties
D) the laws of physics must appear the same to all observers moving with uniform velocity relative to each other
E) everything is relative

Question 23

An event occurs at x = 500 m, t = 0.90 $\mu$ s in one frame of reference. Another frame is moving at 0.90c in the negative x direction. The origins coincide at t = 0 and clocks in the second frame are zeroed when the origins coincide. The coordinate and time of the event in the second frame is:

A) 500 m, 0.90 $\mu$ s
B) 1700 m, 5.5 $\mu$ s
C) 740 m, 2.4 $\mu$ s
D) 260 m, -0.60 $\mu$ s
E) 590 m, -1.4 $\mu$ s

Question 24

Two events occur on the x axis separated in time by $\Delta$ t and in space by $\Delta$ x. A reference frame, traveling at less than the speed of light, in which the two events occur at the same time:

A) exists no matter what the values of $\Delta$ x and $\Delta$ t
B) exists only if $\Delta$ x/ $\Delta$ t < c
C) exists only if $\Delta$ x/ $\Delta$ t > c
D) exists only if $\Delta$ x/ $\Delta$ t = c
E) does not exist under any condition

Question 25

While emitting light of proper frequency f₀, a source moves to the right with speed c/4 relative to reference frame S. A detector, to the left of the source, measures the frequency to be f, which is greater than f₀. This means:

A) the detector is moving to the right with a speed that is greater than c/4 relative to S
B) the detector is moving to the right with a speed that is less than c/4 relative to S
C) the detector is moving to the left with a speed that is greater than c/4 relative to S
D) the detector is moving to the left with a speed that is less than c/4 relative to S
E) the detector is not moving

Question 26

Two flashes of light occur simultaneously at t = 0 in reference frame S, one at x = 0 and the other at x = 600 m. They are observed in reference frame S', which is moving at 0.95c in the positive x direction. The origins of the two frames coincide at t = 0 and the clocks of S' are zeroed when the origins coincide. In S' the coordinate where the leading edges of the two light flashes meet and the time when they meet are:

A) 300 m, 1.0 $\mu$ s
B) 15 m, 0.050 $\mu$ s
C) 585 m, 1.95 $\mu$ s
D) 48 m, 0.16 $\mu$ s
E) 1900 m, 0.16 $\mu$ s

Question 27

Two events occur on the x axis separated in time by $\Delta$ t and in space by $\Delta$ x. A reference frame, traveling at less than the speed of light, in which the two events occur at the same coordinate:

A) exists no matter what the values of $\Delta$ x and $\Delta$ t
B) exists only if $\Delta$ x/ $\Delta$ t < c
C) exists only if $\Delta$ x/ $\Delta$ t > c
D) exists only if $\Delta$ x/ $\Delta$ t = c
E) does not exist under any condition

Question 28

A consequence of Einstein's theory of relativity is:

A) moving clocks run more slowly than when they are at rest
B) moving rods are longer than when they are at rest
C) light has both wave and particle properties
D) the laws of physics must appear the same to all observers moving with uniform velocity relative to each other
E) everything is relative

Question 29

Two electrons move in opposite directions at 0.70c as measured in the laboratory. The speed of one electron as measured from the other is:

A) 0.35c
B) 0.70c
C) 0.94c
D) 1.00c
E) 1.40c

Question 30

An event occurs at x = 500 m, t = 0.90 $\mu$ s in one frame of reference. Another frame is moving at 0.90c in the positive x direction. The origins coincide at t = 0 and clocks in the second frame are zeroed when the origins coincide. The coordinate and time of the event in the second frame is:

A) 500 m, 0.90 $\mu$ s
B) 1700 m, 5.5 $\mu$ s
C) 740 m, 2.4 $\mu$ s
D) 260 m, -0.60 $\mu$ s
E) 590 m, -1.4 $\mu$ s

Question 31

Frame S' moves in the positive x direction at 0.6c with respect to frame S. A particle moves in the positive x direction at 0.4c as measured by an observer in S'. The speed of the particle as measured by an observer in S is:

A) c/5
B) 5c/19
C) 8c/25
D) 25c/31
E) c

Question 32

Relative to reference frame 1, reference frame 2 moves with speed v in the negative x direction. When the origins of the two frames coincide the clocks in both frames are set to zero. An event occurs at coordinate x₁ and time t₁ as measured in reference frame 1 and at coordinate x₂ and time t₂ as measured in frame 2. If , then the coordinates and times of the event are related by:

A) x₂ = $\gamma$ [x₁ - vt₁] and t₂ = $\gamma$ [t₁ - vx₁ / c²]
B) x₂ = $\gamma$ [x₁ - vt₁] and t₂ = $\gamma$ [t₁ + vx₁ / c²]
C) x₂ = $\gamma$ [x₁ + vt₁] and t₂ = $\gamma$ [t₁ - vx₁ / c²]
D) x₂ = $\gamma$ [x₁ + vt₁] and t₂ = $\gamma$ [t₁ + vx₁ / c²]
E) none of the above are correct

Question 33

As a rocket ship moves by at 0.95c a mark is made on a stationary axis at the front end of the rocket and 9 * 10^-8 s later a mark is made on the axis at the back end. The marks are found to be 100 m apart. The rest length of the rocket is:

A) 31 m
B) 78 m
C) 100 m
D) 240 m
E) 320 m

Question 34

Light from some stars shows an apparent change in frequency because of:

A) interference
B) refraction by layers of air
C) diffraction
D) reflection
E) relative motion

Question 35

Two events occur 100 m apart with an intervening time interval of 0.37 $\mu$ s. The speed of a clock that measures the proper time between the events is:

A) 0
B) 0.45c
C) 0.56c
D) 0.90c
E) 1.8c

Question 36

A rocket traveling with constant velocity makes a 8.4 * 10¹⁵ m trip in 1 year. The proper time in years between events which mark the beginning and end of the trip is:

A) 0.21
B) 0.46
C) 1.0
D) 2.2
E) 4.7

Question 37

According to the theory of relativity:

A) moving clocks run fast
B) energy is not conserved in high speed collisions
C) the speed of light must be measured relative to the ether
D) momentum is not conserved in high speed collisions
E) none of the above

Question 38

Observer A measures the velocity of a rocket as and a comet as . Here are parallel and in the direction of the observer's positive x axis. The speed of the comet as measured by an observer on the rocket is:

A) (u - v)/(1 - uv/c²)
B) (u - v)/(1 - v²/c²)
C) (u - v)/(1 - v²/c²)^1/2
D) (u - v)/(1 + uv/c²)
E) (u + v)/(1 - uv/c²)

Question 39

Star S1 is moving away from us at a speed of 0.8c. Star S2 is moving away from us in the opposite direction at a speed of 0.5c. The speed of S1 as measured by an observer on S2 is:

A) 0.21c
B) 0.5c
C) 0.93c
D) 1.3c
E) 2.17c

Question 40

Two events occur 100 m apart with an intervening time interval of 0.60 $\mu$ s. The speed of a reference frame in which they occur at the same coordinate is:

A) 0
B) 0.25c
C) 0.56c
D) 1.1c
E) 1.8c

Question 41

An electron (m = 9.11 *10^-31 kg) has a speed of 0.95c. The magnitude of its momentum is:

A) 2.6 * 10^-22 kg .m/s
B) 2.9 *10^-22 kg .m/s
C) 6.0 *10^-22 kg . m/s
D) 8.3 *10^-22 kg .m/s
E) 8.8* 10^-22 kg .m/s

Question 42

If the kinetic energy of a particle is equal to its rest energy then its speed must be:

A) 0.25c
B) 0.50c
C) 0.87c
D) c
E) unknown unless its mass is given

Question 43

A source at rest emits light of wavelength 500 nm. When it is moving at 0.90c away from an observer, the observer detects light of wavelength:

A) 26 nm
B) 115 nm
C) 500 nm
D) 2200 nm
E) 9500 nm

Question 44

Visible light, with a frequency of 6.0 *10¹⁴ Hz, is reflected from a spaceship moving directly away at a speed of 0.90c. The frequency of the reflected waves observed at the source is:

A) 3.2 * 10¹³ Hz
B) 1.4 * 10¹⁴ Hz
C) 6.0 * 10¹⁴ Hz
D) 2.6 *10¹⁵ Hz
E) 1.1 * 10¹⁶ Hz

Question 45

An electron is moving at 0.6c. If we calculate its kinetic energy using (1/2)mv², we get a result which is:

A) just right
B) just half enough
C) twice the correct value
D) about 1% too low
E) about 28% too low

Question 46

A console lamp in the cabin of a spaceship appears green when the ship and observer are both at rest. When the ship is moving at 0.90c away from Earth, passengers on board see:

A) a dark lamp (the frequency is too high to be seen)
B) a dark lamp (the frequency is too low to be seen)
C) a red lamp
D) a violet lamp
E) a green lamp

Question 47

A particle with zero mass and energy E carries momentum:

A) Ec
B) Ec²
C)
D) E/c
E) E/c²

Question 48

Light from a stationary spaceship is observed, then the spaceship moves directly away from the observer at high speed. As a result, the light seen by the observer has:

A) a higher frequency and a longer wavelength than before
B) a lower frequency and a shorter wavelength than before
C) a higher frequency and a shorter wavelength than before
D) a lower frequency and a longer wavelength than before
E) the same frequency and wavelength as before

Question 49

If the kinetic energy of a free particle is much greater than its rest energy then its kinetic energy is proportional to:

A) the magnitude of its momentum
B) the square of the magnitude of its momentum
C) the square root of the magnitude of its momentum
D) the reciprocal of the magnitude of its momentum
E) none of the above

Question 50

If the mass of a particle is zero its speed must be:

A) c
B) infinite
C) 0
D) any speed less than c
E) any speed greater than c

Question 51

A particle with rest mass m moves with speed 0.6c. Its kinetic energy is:

A) 0.18mc²
B) 0.22mc²
C) 0.25mc²
D) mc²
E) 1.25mc²

Question 52

An electron (m = 9.11 *10^-31 kg) has a speed of 0.95c. Its kinetic energy is:

A) 8.2 * 10^-14 J
B) 1.8 *10^-13 J
C) 2.0 *10^-13 J
D) 2.2 *10^-13 J
E) 2.6 *10^-13 J

Question 53

The work that must be done to increase the speed of an electron (m = 9.11*10^-31 kg) from 0.90c to 0.95c is:

A) 2.6 *10^-13 J
B) 8.2 * 10^-13 J
C) 3.2* 10^-13 J
D) 7.4 * 10^-14 J
E) 3.8*10^-15 J

Question 54

A train traveling very fast (v = 0.6c) has an engineer (E) at the front, a guard (G) at the rear and an passenger (S') exactly half way between them. Both E and G are equipped with yellow signaling lamps. The train passes a station, closely observed by the station master (S). Both E and G use their lamps to send signals. According to both S and S' these signals arrive simultaneously at the instant S' is passing S. According to S, the signal from E will look ______ and that from G will look _____:

A) red, blue
B) yellow, yellow
C) blue, red
D) blue, blue
E) red, red

Question 55

The velocity of an electron is changed from c/2 in the -x direction to c/2 in the +x direction. As a result, its kinetic energy changes by:

A) mc²
B) 0.5mc²
C)
D) 2mc²
E) zero

Question 56

A spectral line of a certain star is observed to be "red shifted" from a wavelength of 500 nm to a wavelength of 1500 nm. Interpreting this as a Doppler effect, the speed of recession of this star is:

A) 0.33c
B) 0.50c
C) 0.71c
D) 0.8c
E) c

Question 57

According to relativity theory a particle of mass m with a momentum of 2mc has a speed of:

A) 2c
B) 4c
C) c
D) c/2
E) 0.89c

Question 58

The mass of a particle is m. In order for its total energy to be twice its rest energy, its momentum must be:

A) mc/2
B)
C) mc
D)
E) 2mc

Question 59

A source at rest emits light of wavelength 500 nm. When it is moving at 0.90c toward an observer, the observer detects light of wavelength:

A) 26 nm
B) 115 nm
C) 500 nm
D) 2200 nm
E) 9500 nm

Question 60

How fast should you move away from a 6.0 * 10¹⁴ Hz light source to observe waves with a frequency of 4.0* 10¹⁴ Hz?

A) 0.20c
B) 0.39c
C) 0.45c
D) 0.51c
E) 0.76c

Question 61

According to the theory of relativity:

A) all forms of energy have mass-like properties
B) moving particles lose mass
C) momentum is not conserved in high speed collisions
D) a rod moving rapidly sideways is shorter along its length
E) a rod moving rapidly sideways is longer along its length

Question 62

A certain particle has a kinetic energy of 3.2 *10^-10 J and a momentum of 1.7 * 10^-18 kg . m/s. Its mass is:

A) 9.11*10^-31 kg
B) 2.7 *10^-27 kg
C) 4.5*10^-27 kg
D) 6.3 * 10^-27 kg
E) 8.6 *10^-27 kg

Question 63

The magnitude of the momentum of a particle can never exceed:

A) mc, where m is its mass
B) E/c, where E is its energy
C) K/c, where K is its kinetic energy
D) none of the above, but there is an upper limit
E) none of the above; there is no upper limit

Question 64

If the kinetic energy of a free particle is much less than its rest energy then its kinetic energy is proportional to:

A) the magnitude of its momentum
B) the square of the magnitude of its momentum
C) the square root of the magnitude of its momentum
D) the reciprocal of the magnitude of its momentum
E) none of the above

Question 65

An electron (m = 9.11 *10^-31 kg) has a momentum of 4.0 * 10^-13 kg .m/s. Its kinetic energy is:

A) 6.3 *10^-14 J
B) 8.2 *10^-14 J
C) 1.2 * 10^-13 J
D) 1.5 *10^-13 J
E) 2.7* 10^-13 J