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5. Relativistic time dilation and length contraction (Ch. 37)
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5. Relativistic time dilation and length contraction (Ch. 37)
Due: 11:59pm on Sunday, February 28, 2016
To understand how points are awarded, read the Grading Policy for this assignment.
Postulates of Special Relativity
Description: Discuss the postulates of Special Relativity, and show that they contradict Galilean Relativity.
Learning Goal:
To understand the postulates of special relativity and their relationship to Galilean relativity.
Einstein's theory of special relativity is based on two postulates:
1. The principle of relativity: The laws of physics are the same in any inertial coordinate system. For example, by watching the action of the balls on a pool
table that is on a ship, you cannot tell whether the ship is at the dock or moving through the water at a constant speed. (You can think of an inertial coordinate
system as a nonaccelerating coordinate system. There are actually other subtle conditions, but for now they are not of concern.)
2. The speed of light in vacuum is constant. This says that observers in any inertial coordinate system will measure the same value for the speed of light,
independent of the origin of that light.
The simplicity of these assumptions belies their brilliance--they directly contradict our intuitive ideas of relativity, yet by accepting them we can easily build a theory of relative
motion that is in accord with all observation.
We now discuss these ideas more fully, showing where they depart from the previously held ideas about relativity.
Part A
Consider a pool game being played on a pool table on the deck of an aircraft carrier near the bow (front). Assume that the carrier is moving north at 25.0
. As a result
of the initial break, a ball flies over the edge of the table and over the edge of the deck with a horizontal component of velocity directed toward the bow of the ship of 5.00
. What is
, the horizontal component of the speed with which this ball strikes the water in front of the ship (i.e., what is the speed of the ball relative to the water)?
Express your answer numerically in meters per second. Use ordinary (Galilean) physics.
Hint 1. Galilean relativity for velocity addition
In Galilean relativity, time and distance are absolute and independent quantities, and velocities therefore add. In particular, the formula for relative velocity addition
is (suppressing the vector signs)
, where
is the velocity of the ball relative to the water,
is the velocity of the ball relative to the carrier,
and
is the velocity of the carrier relative to the water.
ANSWER:
= 30
Part B
You have used the ideas of Galilean relativity--that time and distance are absolute and independent quantities and that velocities therefore add. In particular, the formula
for relative velocity addition is
, where
is the velocity of the ball relative to the water,
is the velocity of the ball relative to the carrier, and
is
the velocity of the carrier relative to the water. These ideas are embodied in Isaac Newton's Principia.
Newton started this seminal work by stating which of the following?
ANSWER:
Space is uniform and infinite in extent but time is relative.
Space is relative but time is everywhere uniform and the same.
Space is uniform and infinite in extent and time is everywhere uniform and the same.
Space and time are both relative.
Part C
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Which of these factors was definitely not a consideration in Einstein's making the postulate that the speed of light is constant for all observers?
ANSWER:
Maxwell's equations predict that light travels at speed
without any reference to the velocity of the reference frame.
The Michelson-Morley experiment showed that the round-trip travel time of light did not depend on the motion of the earth through the ether.
Measurements of the locations of distant stars showed an annual circular motion consistent with the hypothesis that the earth was moving through a stationary
ether in which the star's light wave traveled.
Michelson made very accurate measurements of the speed of light.
The exact value of the speed of light was immaterial to Einstein except that it had to be a speed beyond practical comprehension. Also, Michelson measured the
speed of light after Einstein's work, which was published in 1905.
Part D
We now consider the motion of a flash of light that is emitted from the ship described in Part A. Imagine that there is a flashlight on the pool table that emits a flash of light
directed toward the bow of the ship, which is still traveling northward at 25.0
. The light encounters two photodiodes on the table that are spaced exactly 1.00
apart
along a north-south axis. What is the time
that elapses between when the flash of light encounters the first photodiode and when it strikes the second?
Express your answer in seconds to eight significant figures. Remember that an answer such as 1/300000025 will be accepted at its calculated value (if you do
not put commas in your answer). Also, the speed of light in vacuum is
.
Hint 1. How to get time from velocity
As always, you can find the time
from the equation
, where
is the distance and
is the velocity.
ANSWER:
= 3.3356410×10−9
Part E
Now imagine that there is a similar set of photodiodes, also spaced 1.00
apart along a north-south axis, mounted on a dock on shore directly ahead of the aircraft
carrier. The light from the flashlight on the pool table encounters these two photodiodes. What is the time
that elapses between when the flash of light encounters
the first photodiode and when it strikes the second?
Express your answer in seconds to eight significant figures. Remember that an answer such as 1/300000025 will be accepted at its calculated value (if you do
not put commas in your answer). Use
for the speed of light in vacuum.
Hint 1. How to get time from velocity
As always, you can find the time
from the equation
, where
is the distance and
is the velocity.
ANSWER:
= 3.3356410×10−9
Part F
You have now correctly used the key ideas of special relativity. As a consequence of these ideas, what can you conclude?
ANSWER:
Space is uniform and infinite in extent but time is relative.
Space is relative but time is everywhere uniform and the same.
Space is uniform and infinite in extent and time is everywhere uniform and the same.
Space and time are both relative.
Gedanken Conceptual Questions
Description: Conceptual questions on observations in different frames of reference. These questions ask students to perform thought experiments similar to Einstein's
gedanken experiments.
Einstein developed much of his understanding of relativity through the use of gedanken, or thought, experiments. In a gedanken experiment, Einstein would imagine an
experiment that could not be performed because of technological limitations, and so he would perform the experiment in his head. By analyzing the results of these
experiments, he was led to a deeper understanding of his theory.
In each the following gedanken experiments, Albert is in the exact center of a glass-sided freight car speeding to the right at a very high speed
relative to you.
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Albert has a flashlight in each hand and directs them at the front and rear ends of the freight car. Albert switches the flashlights on at the same time.
Part A
In Albert's frame of reference, which beam of light travels at a greater speed, the one directed toward the front or the one toward the rear of the train, or do they travel at
the same speed? Which beam travels faster in your frame of reference?
Enter the answers for Albert's frame of reference and your frame of reference separated by a comma using the terms front, rear, and same. For example, if in
Albert's frame of reference the beam of light directed toward the front of the train travels at a greater speed and in your frame of reference the two beams
travel at the same speed, then enter front,same.
Hint 1. A postulate of Einstein’s relativity
One of the basic postulates of special relativity is that the speed of light is the same for all observers, regardless of the speed of either the observer or the source
of the light.
ANSWER:
Part B
In Albert's frame of reference, which end, front or rear, is struck by light first, or are they struck at the same time? Which end is struck first in your frame of reference?
Enter the answer for Albert's frame of reference and your frame of reference separated by a comma using the terms front, rear, and same. For example, if in
Albert's frame of reference the beam of light strikes the front first and in your frame of reference the two beams strike at the same time, then enter
front,same.
Hint 1. Distinguishing between frames of reference
It is important to clearly distinguish between the two frames of reference involved in this problem. In Albert’s frame, the train is at rest. Albert, the flashlights, the
walls, and the ceiling and floor of the train are all completely stationary in this frame of reference. In your frame of reference, the train and everything in it is moving.
Albert, the flashlights, the walls, and the ceiling and floor of the train are all moving at exactly the same speed in your frame of reference.
Hint 2. Situation in Albert’s frame
In Albert’s frame, the train is completely at rest. The light from the flashlights is emitted directly between the two stationary ends, and one beam moves to the right
and the other moves to the left at exactly the same speed, the speed of light.
Hint 3. Situation in your frame
In your frame, both the front and rear ends are moving to the right at the same constant speed. The light from the flashlights is emitted directly between these two
moving ends, and one beam moves to the right and the other moves to the left at exactly the same speed, the speed of light.
ANSWER:
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One of the most startling consequences of special relativity is that two events can be simultaneous for one observer, in one frame of reference, but may not be
simultaneous for a different observer in a different frame. Thus, saying that two things "happened at the same time" may be true for one person and false for another,
but both observers are correct in their statements.
Now Albert directs his flashlights at the ceiling and floor of the freight car. The flashlights are located midway between the ceiling and the floor and Albert switches them on at
the same time
Part C
In Albert's frame of reference, which surface, ceiling or floor, is struck by light first, or are they struck at the same time? Which surface is struck first in your frame of
reference?
Enter the answer for Albert's frame of reference and your frame of reference separated by a comma using the terms ceiling, floor, and same. For example, if in
Albert's frame of reference the ceiling is struck by light first and in your frame of reference the floor and the ceiling are struck by light at the same time, then
enter ceiling,same.
Hint 1. Situation in Albert's frame
In Albert’s frame, the train is completely at rest. The light from the flashlights is emitted directly between the floor and the ceiling, and one beam moves up and the
other moves down at exactly the same speed, the speed of light.
Hint 2. Situation in your frame
In your frame, the entire train is moving to the right, but no part of the train is moving vertically. The light from the flashlights is emitted directly between the floor
and the ceiling, and one beam moves up and the other moves down at exactly the same speed, the speed of light.
ANSWER:
Albert is playing laser tag in the freight car. Two "assassins" sneak into the freight car with Albert. One is positioned against the front end and the other against the rear end.
They each fire a laser at Albert. The two lasers strike Albert at the same time.
Part D
In Albert's frame of reference, who fired first, the person against the rear end or the person against the front end, or did they fire at the same time? In your frame of
reference, who fired first?
Enter the answers for Albert's frame of reference and your frame of reference separated by a comma using the terms front, rear, and same. For example, if in
Albert's frame of reference the person against the front end fired first and in your frame of reference both "assassins" fired at the same time, then enter
front,same.
Hint 1. Situation in Albert's frame
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In Albert’s frame, the train is completely at rest. The two assassins are equally distant from Albert and their laser beams travel at equal speeds.
Hint 2. Situation in your frame
In your frame, Albert is directly between the assassins and moving to the right. The rear laser beam moves to the right at the speed of light and the front laser
beam moves to the left at the speed of light.
Therefore, Albert is moving in the same direction as the rear beam and in the opposite direction of the front beam. Which beam will take a larger amount of time to
hit Albert?
ANSWER:
front
rear
The question states that both beams strike Albert at the same time. Therefore the one that takes a larger amount of time to reach him must have been fired
first.
ANSWER:
This is another example of how two events, the firing of the lasers, can be simultaneous in one frame of reference and not simultaneous in another frame.
Time is Relative
Description: Consider two events occurring at the same space point in a frame of reference, find their space distance as measured in a moving frame of reference, given
the time intervals between the two events measured in both frames of reference.
Part A
Two events are observed in a frame of reference S to occur at the same space point, with the second event occurring after a time of 1.70 . In a second frame S' moving
relative to S, the second event is observed to occur after a time of 2.25 . What is the difference
between the positions of the two events as measured in S'?
Use 3.00×108
for the speed of light in a vacuum.
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Hint 1. How to approach the problem
Although the two events are observed to occur at the same location in S, from the point of view of an observer at rest in S' they occur at a space distance that
depends on the relative speed of the two frames of reference and the time elapsed between the two events as measured by the observer in S'.
Note that the observer at rest in S' will see the source of the event moving with speed . Therefore, to find the difference between the positions of the two events
as measured in S', you need to determine the relative speed . To do that, use the relativistic relationship between the time interval measured in S and the one
measured in S'.
Hint 2. Find the expression for the space distance between the two events
If the speed of S' relative to S is , which of the following expressions for the space distance between the two events as measured in S' is correct?
Here,
and
are the time intervals between the two events as measured in S and S', respectively.
ANSWER:
Hint 3. Find the relative speed
What is the speed
of the frame of reference S ' with respect to the frame S?
Express your answer numerically in meters per second.
Hint 1. Time dilation
Since the speed of light is the same in all frames of reference (Einstein's postulate), the time interval
between two events that are observed to occur at
the same space point in a given frame of reference S is not the same in all reference frames. An observer in a frame S' moving with speed relative to S
would measure a longer time interval between those two same events because of the relativistic effect of time dilation. The time interval
measured in the
moving frame S' is
.
If you know the time interval
, also called proper time, and
, you can derive an expression for the relative speed .
Hint 2. Find the proper time
The time interval between two events occurring at the same space point as measured in a particular frame of reference is called the proper time. What is the
value of the proper time
in this case?
Express your answer numerically in seconds.
ANSWER:
=
= 1.70
ANSWER:
= 1.96×108
=
Also accepted:
= −1.96×108
ANSWER:
=
Also accepted:
= 4.42×108
= −4.42×108
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Do not be tempted to interpret the distance that you have just calculated as the distance traveled by the moving frame S' in the time elapsed between the two events
as measured by an observer in S. An observer in S would measure a distance equal to
, where is the relative speed of the two reference frames and
is the
time between events in S.
Exercise 37.5
Description: The negative pion pi^- is an unstable particle with an average lifetime of 2.60 * 10^-8 s (measured in the rest frame of the pion). (a) If the pion is made to
travel at very high speed relative to a laboratory, its average lifetime is measured in the ...
The negative pion
is an unstable particle with an average lifetime of
(measured in the rest frame of the pion).
Part A
If the pion is made to travel at very high speed relative to a laboratory, its average lifetime is measured in the laboratory to be
pion expressed as a fraction of .
. Calculate the speed of the
Express your answer using five significant figures.
ANSWER:
= 0.99808
Part B
What distance, measured in the laboratory, does the pion travel during its average lifetime?
Express your answer using three significant figures.
ANSWER:
= 126
The Empire Strikes Back
Description: A space ship marked by an elliptic symbol of given dimensions is moving relative to an observer; find the speed of the space ship needed in order for the
observer to measure the symbol as circular.
The starships of the Solar Federation are marked with the symbol of the Federation, a circle, whereas starships of the Denebian Empire are marked with the Empire's symbol,
an ellipse whose major axis is times its minor axis (
in the figure ).
Part A
How fast, relative to an observer, does an Empire ship have to travel for its markings to be confused with those of a Federation ship? Use
vacuum.
Express your answer in terms of
for the speed of light in a
and .
Hint 1. How to approach the problem
An observer would see a circle on the Empire ship if the major axis and the minor axis of the Empire's symbol appear to be of equal length. Using the
relativistic relation for length contraction calculate the length of the major axis of the Empire's symbol as measured by the observer and make it equal to its minor
axis. By doing so you will obtain an expression for the speed of the Empire ship. Note that a length perpendicular to the direction of motion is not affected by the
relativistic effect of length contraction. So the observer at rest relative to the Empire ship will measure the same length for the minor axis of the Empire's symbol as
an observer on the Empire ship.
Hint 2. Find the length of the major axis as measured by the observer
If the Empire ship is moving at speed relative to an observer, what is the length of the major axis
denotes the speed of light and is the proper length of the major axis of the Empire's symbol.
in the Empire's symbol as measured by the observer?
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ANSWER:
Now make this expression equal to the minor axis of the Empire's symbol and solve for the relative speed . Note that the proper length of the minor axis will
cancel out.
ANSWER:
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