Chapter 37 Identify that a Galilean transformation is approximately correct

Chapter: Chapter 37

Learning Objectives

LO 37.1.0 Solve problems related to simultaneity and time dilation.

LO 37.1.1 Identify the two postulates of (special) relativity and the type of frames to which they

apply.

LO 37.1.2 Identify the speed of light as the ultimate speed and give its approximate value.

LO 37.1.3 Explain how the space and time coordinates of an event can be measured with a three

dimensional array of clocks and measuring rods and how that eliminates the need of a signal’s

travel time to an observer.

LO 37.1.4 Identify that the relativity of space and time has to do with transferring measurements

between two inertial frames with relative motion but we still use classical kinematics and

Newtonian mechanics within a frame.

LO 37.1.5 Identify that for reference frames with relative motion, simultaneous events in one of

the frames will generally not be simultaneous in the other frame.

LO 37.1.6 Explain what is meant by the entanglement of the spatial and temporal separations

between two events.

LO 37.1.7 Identify the conditions in which a temporal separation of two events is a proper time.

LO 37.1.8 Identify that if the temporal separation of two events is a proper time as measured in

one frame, that separation is greater (dilated) as measured in another frame.

LO 37.1.9 Apply the relationship between proper time Δt0, dilated time Δt, and the relative speed

v between two frames.

LO 37.1.10 Apply the relationships between the relative speed v, speed parameter β, and the

Lorentz factor γ.

LO 37.2.0 Solve problems related to the relativity of length.

LO 37.2.1 Identify that because spatial and temporal separations are entangled, measurements of

the lengths of objects may be different in two frames with relative motion.

LO 37.2.2 Identify the condition in which a measured length is a proper length.

LO 37.2.3 Identify that if a length is a proper length as measured in one frame, the length is less

(contracted) as measured in another frame that is in relative motion parallel to the length.

LO 37.2.4 Apply the relationship between contracted length L, proper length L0, and the relative

speed v between two frames.

LO 37.3.0 Solve problems related to the Lorentz transformations.

LO 37.3.1 For frames with relative motion, apply the Galilean transformation to transform an

event’s position from one frame to the other.

LO 37.3.2 Identify that a Galilean transformation is approximately correct for slow relative

speeds but the Lorentz transformations are the correct transformations for any physically

possible speed.

LO 37.3.3 Apply the Lorentz transformations for the spatial and temporal separations of two

events as measured in two frames with a relative speed v.

LO 37.3.4 From the Lorentz transformations, derive the equations for time dilation and length

contraction.

LO 37.3.5 From the Lorentz transformations show that if two events are simultaneous but

spatially separated in one frame, they cannot be simultaneous in another frame with relative

motion.

LO 37.4.0 Solve problems related to the relativity of velocities.

LO 37.4.1 With a sketch, explain the arrangement in which a particle’s velocity is to be

measured relative to two frames that have relative motion.

LO 37.4.2 Apply the relationship for a relativistic velocity transformation between two frames

with relative motion.

LO 37.5.0 Solve problems related to Doppler effect for light.

LO 37.5.1 Identify that the frequency of light as measured in a frame attached to the light source

(the rest frame) is the proper frequency.

LO 37.5.2 For source–detector separations increasing and decreasing, identify whether the

detected frequency is shifted up or down from the proper frequency, identify that the shift

increases with an increase in relative speed, and apply the terms blue shift and red shift.

LO 37.5.3 Identify radial speed.

LO 37.5.4 For source–detector separations increasing and decreasing, apply the relationships

between proper frequency f0, detected frequency f, and radial speed v.

LO 37.5.5 Convert between equations for frequency shift and wavelength shift.

LO 37.5.6 When a radial speed is much less than light speed, apply the approximation relating

wavelength shift Δλ, proper wavelength λ0, and radial speed v.

LO 37.5.7 Identify that for light (not sound) there is a shift in the frequency even when the

velocity of the source is perpendicular to the line between the source and the detector, an effect

due to time dilation.

LO 37.5.8 Apply the relationship for the transverse Doppler effect by relating detected frequency

f, proper frequency f0, and relative speed v.

LO 37.6.0 Solve problems related to momentum and energy.

LO 37.6.1 Identify that the classical expressions for momentum and kinetic energy are

approximately correct for slow speeds whereas the relativistic expressions are correct for any

physically possible speed.

LO 37.6.2 Apply the relationship between momentum, mass, and relative speed.

LO 37.6.3 Identify that an object has a mass energy (or rest energy) associated with its mass.

LO 37.6.4 Apply the relationships between total energy, rest energy, kinetic energy, momentum,

mass, speed, the speed parameter and the Lorentz factor.

LO 37.6.5 Sketch a graph of kinetic energy versus the ratio v/c (of speed to light speed) for both

classical and relativistic expressions of kinetic energy.

LO 37.6.6 Apply the work–kinetic energy theorem to relate work by an applied force and the

resulting change in kinetic energy.

LO 37.6.7 For a reaction, apply the relationship between the Q value and the change in the mass

energy.

LO 37.6.8 For a reaction, identify the correlation between the algebraic sign of Q and whether

energy is released or absorbed by the reaction.

Multiple Choice

1. 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

2. 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

3. 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

4. The speed of light in vacuum is approximately

A) 186,000 miles per hour

B) 300,000 km per minute

C) one foot per nanosecond

D) 186,000 feet per second

E) 300,000 meters per second

5. 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 at 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

6. 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 on the Earth’s surface

E) none of the above

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

8. As we watch, a spaceship passes us in time t. The crew of the spaceship measures the

passage time and finds it to be t‘. Which of the following statements is true?

A) t is the proper time for the passage and it is smaller than t‘

B) t is the proper time for the passage and it is greater than t‘

C) t‘ is the proper time for the passage and it is smaller than t

D) t‘ is the proper time for the passage and it is greater than t

E) None of the above statements are true.

9. A millionairess was told in 1992 that she had exactly 15 years to live. However, if she

immediately takes off, 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) 2017

E) 2033

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

the clock is:

A) 0.0100c

B) 0.100c

C) 0.900c

D) 0.990c

E) 0.995c

11. A meson when at rest decays 2 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) 4.6 s

D) 14 s

E) none of these

12. Pi mesons at rest have a half-life of T. If a beam of pi mesons is traveling at a speed of

v =



c, the distance in which the intensity of the beam is halved is:

A) c



T(1 –



2)–1/2

B) c



T[(1 +



)/(1 –



)]1/2

C)



vT

D) (1 –



2)1/2vT

E) none of the above

13. 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

14. A meter stick moves sideways (that is, in a direction perpendicular to its length) at 0.95c.

According to measurements taken in the laboratory, its length is:

A) 0 m

B) 0.098 m

C) 0.31 m

D) 1.0 m

E) 3.2 m

15. A consequence of Einstein’s theory of relativity is:

A) moving clocks appear to run more slowly than when they are at rest

B) moving rods appear 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

16. 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

17. 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

18. A consequence of Einstein’s theory of relativity is:

A) moving clocks appear to run faster than when they are at rest

B) moving rods appear 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

19. 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

20. 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 s

B) 0.25 s

C) 0.33 s

D) 0.52 s

E) 0.69 s

21. A certain automobile is 6.0 m long if at rest. If it is measured to be 4.8 m long while

moving, its speed is:

A) 0.1c

B) 0.3c

C) 0.6c

D) 0.8c

E) > 0.95c

22. 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 s

B) 0.80 s

C) 1.00 s

D) 1.25 s

E) 1.67 s

23. Two events occur on the x axis separated in time by t and in space by 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 x and t

B) exists only if x/t < c

C) exists only if x/t > c

D) exists only if x/t = c

E) does not exist under any condition

24. Two events occur on the x axis separated in time by t and in space by 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 x and t

B) exists only if x/t < c

C) exists only if x/t > c

D) exists only if x/t = c

E) does not exist under any condition

25. Which statement is correct?

A) Galilean transformations are correct at any relative speed, but Lorentz transformations are

only approximately correct for relative speeds near the speed of light.

B) Lorentz transformations are correct at any relative speed, but Galilean transformations are

only approximately correct for relative speeds near the speed of light.

C) Galilean transformations are correct at any relative speed, but Lorentz transformations are

only approximately correct for relative speeds that are small compared to the speed of light.

D) Lorentz transformations are correct at any relative speed, but Galilean transformations are

only approximately correct for relative speeds that are small compared to the speed of light.

E) Galilean transformations are only approximately correct for relative speeds that are small

compared to the speed of light, and Lorentz transformations are only approximately correct for

relative speeds near the speed of light.

26. The length of a meter stick moving at 0.95c in the direction of its length with respect to the

laboratory is measured by simultaneously marking its ends on an axis which is stationary in the

laboratory. 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 s

B) 1.1  10–9 s

C) 3.2  10–9 s

D) 3.5  10–9 s

E) 1.1  10–8 s

27. Two events occur 100 m apart with an intervening time interval of 0.60 s. The speed of a

reference frame in which they occur at the same coordinate is:

A) 0 c

B) 0.25c

C) 0.56c

D) 1.1c

E) 1.8c

28. Two independent events occur 100 m apart with an intervening time interval of 0.42 s.

The proper time between the events is:

A) 0 s

B) 0.16 s

C) 0.26 s

D) 0.42 s

E) 0.69 s

29. Two events occur 100 m apart with an intervening time interval of 0.37 s. The speed of a

clock that measures the proper time between the events is:

A) 0 c

B) 0.45c

C) 0.56c

D) 0.90c

E) 1.8c

30. A rocket traveling with constant velocity makes an 8.4  1015 m trip in 1 year. The proper

time between events which mark the beginning and end of the trip is:

A) 0.21 years

B) 0.46 years

C) 1.0 years

D) 2.2 years

E) 4.7 years

31. 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 9.0  10–8 s later a mark is made on the axis at the back end. The marks are 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

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 𝛾 = 1/√1 − 𝑣2/𝑐2, then the coordinates and times of

the event are related by:

A) x2 =  [x1 – vt1] and t2 =  [t1 – vx1 / c2]

B) x2 =  [x1 – vt1] and t2 =  [t1 + vx1 / c2]

C) x2 =  [x1 + vt1] and t2 =  [t1 – vx1 / c2]

D) x2 =  [x1 + vt1] and t2 =  [t1 + vx1 / c2]

E) none of the above are correct

33. An event occurs at x = 500 m, t = 0.90 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 s

B) 1700 m, 5.5 s

C) 740 m, 2.4 s

D) 260 m, –0.60 s

E) 590 m, –1.4 s

34. An event occurs at x = 500 m, t = 0.90 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 s

B) 1700 m, 5.5 s