Physics 2020, Spring 2005
Lab 10
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Lab 10: Relativity
INTRODUCTION & BACKGROUND:
We have recently spent some time studying electromagnetic waves, which include
radio waves, microwaves, light waves, etc. As the understanding of electromagnetic
wave behavior was being developed in the 19th century, one issue continued to nag
at the scientific community – what was actually waving? For example, with water
waves we know that the water is oscillating, and for sound waves, we know that the
air is oscillating. For electromagnetic waves, what is the “waving” medium? One
popular idea, which dated back to the days of Aristotle (though it had never been
observed), was that the universe was filled with “ether”, which the planets and stars
moved through.
A famous experiment to test how fast the earth was moving through this ether was
the Michelson-Morley experiment, which we discussed in class. Interestingly, the
experiment showed that there was no difference in the speed of light regardless of
which direction the light source was moving relative to the wave propagation. While
the experimental results were controversial for many years, Einstein was eventually
able to propose a framework in which the results could be trusted, but which required
some modifications to some of the most basic tenets of Newtonian physics.
As discussed in class, the ideas of special relativity are largely consequences of the
following two simple premises:
1. The laws of physics are the same in all inertial reference frames.
2. The speed of light in vacuum is a constant (c = 299792458 m/s)
and is independent of the motion of its source.
It is generally this second premise that is the most counterintuitive – we know, for
example that the speed of sound waves definitely is dependent on the motion of their
source (we can hear this every time an ambulance passes by and the siren is
Doppler shifted). Not so with light – if a person in the car shines a light backwards
out of the back of the car, both the person in the car and the person on the street will
measure the light going at exactly 299792458 m/s. Experimentally, this has been
proven to be true in countless experiments since Einstein.
During this lab we will see that most relativistic effects are extremely small unless the
objects involved are moving close to the speed of light c, so it is often simpler to refer
to the velocity in terms of a fraction of the speed of light. Because relativity
experiments involve such tremendous speeds, this lab will consist of a series of
“thought experiments” to get at the basic ideas of special relativity.
University of Colorado at Boulder, Department of Physics
Physics 2020, Spring 2005
Lab 10
page 2 of 10
PART I: Time Dilation
Consider a baseball pitcher riding in his wife’s convertible on his way to the next
game. He is capable of pitching the ball at 40 m/s (about 90 mph), and the car is
moving at 30 m/s (about 65 mph). If he throws a baseball forward out of the car,
how fast does the pitcher (riding in the car) measure it moving as it leaves his hand?
The ball goes out the front of the car and is caught by the team catcher, who
happened to be hanging out by the side of the highway. Just before he caught the
ball, how fast did the catcher measure the ball moving?
Consider the same situation, but now the pitcher throws the ball backwards out of the
car just after he passes the catcher. How fast does the pitcher measure the ball to
be going? How about the catcher?
Consider a similar experiment, but now the pitcher is riding along at night, and turns
on a flashlight, aimed forward. How fast does the pitcher (in the car) measure the
light to be going?
How fast does the catcher, standing by the side of the road, measure the light to be
going?
What happens if the flashlight is instead aimed backwards out of the car?
University of Colorado at Boulder, Department of Physics
Physics 2020, Spring 2005
Lab 10
page 3 of 10
Consider a new kind of time-measuring device, called a “light
clock”. The light-clock consists of a transmitter (flash bulb) and a
mirror separated by a distance L, and a receiver (photodetector)
located next to the transmitter. The transmitter sends out a pulse
of light, which bounces off the mirror and returns to the receiver.
Each time the receiver receives a light pulse, it triggers the
transmitter to send out another pulse, so that it “ticks” along with
a constant period between each pulse. A stopwatch is placed
next to the transmitter and receiver to measure the time interval
between each clock “tick”.
How far does the light travel in each round-trip from the
transmitter to the receiver?
What is the time interval ∆t between each light pulse?
Now consider what happens if the whole lightclock is moving to the right at some high
speed v. Light is sent to the mirror and
bounced back to the receiver, which moved
while the light was traveling. In order to make
extremely precise measurements, a second
stopwatch (perfectly synchronized with the
first one) is placed near where the light
arrives, to measure the time of its arrival.
How far in the x-direction does the lightclock move in the time interval ∆t’ that it takes
the light to make the round-trip from the
transmitter to the receiver?
(Write your
answer in terms of v and ∆t’)
How far in the y-direction does the light travel in each round-trip from the transmitter
to the receiver?
What is the total distance that the light travels in each round-trip from the transmitter
to the receiver?
University of Colorado at Boulder, Department of Physics
Physics 2020, Spring 2005
Lab 10
page 4 of 10
Noting that distance = rate × time, what is the time interval ∆t’ between each light
pulse? (Notice the subtle change in notation – the “primed” value is the value which
is moving relative to the observer.)
How does the time interval ∆t’ (for the stationary observer measuring the moving
clock) compare with ∆t (for the stationary observer watching the stationary clock)?
Specifically, what is ∆t’ / ∆t ?
Now consider a person moving with the light-clock. According to that person, how
far must the light travel in each round-trip from the transmitter to the receiver?
According to the person moving with the light-clock, what is the time interval ∆t
between each light pulse?
What appears to be happening to the rate of the light-clock when it is moving relative
to the observer? Be precise.
University of Colorado at Boulder, Department of Physics
Physics 2020, Spring 2005
Lab 10
page 5 of 10
Part II. Using the Light-Clock to Measure Length (with algebra!)
Let’s do the light-clock thought experiment
again, but this time aiming the clock along the
direction of motion.
First, just to review, consider the light-clock at rest with respect to the observer. How
far does the light travel in each round-trip from the transmitter to the receiver?
What is the time interval ∆t between each clock tick?
Now consider what happens if the
whole light-clock is moving to the
right at some high speed v. Light
is sent to the mirror and bounced
back to the receiver, which
moved while the light was
traveling.
Before we go too far, let’s write down all the variables that we’re going to use, to help
us keep them straight: (primed variables denote quantities in a moving frame,
unprimed variables denote quantities in a non-moving frame)
L
length of the light-clock in non-moving frame
L’
length of the light-clock in moving frame
∆t
time interval for light to travel one complete round-trip in non-moving frame
∆t1’
time interval for light to travel from transmitter to mirror in moving frame
∆t2’
time interval for light to travel from mirror to receiver in moving frame
∆t’
time interval for light to travel one complete round-trip in moving frame
University of Colorado at Boulder, Department of Physics
Physics 2020, Spring 2005
Lab 10
page 6 of 10
How far does the mirror move in the time interval ∆t1’ that it takes the light to travel
from the transmitter to the mirror? (Write your answer in terms of v and ∆t1’)
How far does the light travel in the time interval ∆t1’ that it takes the light to travel
from the transmitter to the mirror? (Write your answer in terms of L’, v and ∆t1’)
Knowing that light travels at speed c, we know that the distance that the light travels
in ∆t1’ can also be written as c×∆t1’. This relationship (i.e. writing the distance that
the light traveled from the transmitter to the mirror in two ways) will be useful in a few
steps, so write it again here:
Distance light travels on first leg = c×∆t1’ = ____________ (eq. 1)
How far does the mirror move in the time interval ∆t2’ that it takes the light to travel
from the mirror to the receiver? (Write your answer in terms of v and ∆t2’)
How far does the light travel in the time interval ∆t2’ that it takes the light to travel
from the mirror to the receiver? (Write your answer in terms of L’, v and ∆t2’)
Knowing that light travels at speed c, we know that the distance that the light travels
in ∆t2’ can also be written as c×∆t2’. This relationship will be useful in a few steps, so
write it again here:
Distance light travels on second leg = c×∆t2’ = ____________ (eq. 2)
Rearranging equations 1 and 2 should give us the following two equations
∆t1 ' =
L'
c+v
∆t 2 ' =
L'
c−v
University of Colorado at Boulder, Department of Physics
Physics 2020, Spring 2005
Lab 10
page 7 of 10
Now, plugging in the previous two equations into the definition of ∆t’ = ∆t1’ + ∆t2’ , we
can write:
1 ⎞
2 L' c
2 L'
⎛ 1
∆t ' = L' ⎜
+
=
⎟= 2
2
⎛ v2
⎝c+v c−v⎠ c −v
c⎜⎜1 − 2
⎝ c
⎞
⎟⎟
⎠
BUT we already know from Part I that the time intervals in the moving frame are
different than those in the non-moving frame by the relationship
∆t ' =
∆t
(verify that this is true)
v2
1− 2
c
AND we also know that for the non-moving clock, a single round-trip for the light
pulse takes an interval
∆t =
2L
c
(verify that this is true)
Putting this all together, we get
∆t ' =
∆t
v2
1− 2
c
=
2L
v2
c 1− 2
c
=
2 L'
⎛ v2
c⎜⎜1 − 2
⎝ c
⎞
⎟⎟
⎠
From this, how does L’ (the length of the clock in the moving frame) compare to L
(the length of the clock in the non-moving frame)? Describe both as an equation and
in words.
University of Colorado at Boulder, Department of Physics
Physics 2020, Spring 2005
Lab 10
page 8 of 10
Part III. The Great Train Robbery
Suppose a 150 meter long train (as measured when not moving) is moving along a
straight track through the mountains at a speed of 0.6 c. On this particular stretch of
track is a 140 meter long tunnel, with doors at both ends which can be
simultaneously (in the reference frame of the track) closed by a master switch. Some
train robbers are planning a robbery, and they want to check whether they can trap
the train in the tunnel. They are still in the planning stages, so they don’t want to
actually disturb the passengers – so their plan is to close the doors and quickly
opening them again before anyone notices.
Ordinarily, of course, a 150 meter train would never fit into a 140 meter tunnel.
v2
(where L’ is the shortened length of a relativisticallyc2
moving object and L is its “proper length” when measured at rest), how long is the
train as measured by the robbers standing next to the track?
Recalling that L' = L 1 −
If the robbers are very careful with the timing of the door-switch, can they trap the
train in the tunnel for a moment?
But wait a minute… What about looking at this problem from the point of view of the
people on the train? According to the people on the train, how long is the train?
According to the people on the train, how long is the tunnel?
Let’s ask again -- does the train get trapped in the tunnel? We learned in class that
two events that are simultaneous in one reference frame are not necessarily
simultaneous in a different reference frame (especially when one reference frame is
moving at relativistic speeds with respect to the other). Can you think of a way that
this can be used to re-describe the sequence of events and explain the paradox?
University of Colorado at Boulder, Department of Physics
Physics 2020, Spring 2005
Lab 10
page 9 of 10
PRELAB QUESTIONS: (to be turned in upon arriving at lab)
1. Read the lab thoroughly, and familiarize yourself with chapter 26.
2. We discussed in class that two twins will age at different rates if one of them is
sent at relativistic speeds to a star and back. The formula we used was
∆t
, where ∆t’ was the age of the person on earth and ∆t was the age
∆t ' =
2
v
1−
c2
of the traveling twin. In our example in class, one twin was sent to a star 10 lightyears away traveling at 0.9c, and aged 9.7 years while her earth-bound sister
aged 22.2 years. Suppose instead you send your twin to Kansas City to get some
barbecue for the NCAA basketball tournament party. She travels the 1900 km
round trip in 20 hours. When she gets back, how much less has she aged than
you?
3. Imagine you are flying through space at a constant speed, in a ship with only one
viewing portal, which is jammed shut (i.e. you can’t look out). Suddenly you
realize that you have lost track of how fast you were going. Is there any
experiment that you can do to know your speed?
4. After a while you are able to open the viewing portal, and you see another space
ship pass you by. It occurs to you that you don’t even know whether you are
moving past a stationary ship, or whether you are standing still, getting passed by
a moving ship. Is there any way you can tell this?
5. Is the earth moving around the sun, or is the sun (along with the rest of the solar
system) moving around the earth? Is there a difference? Is there a way to test?
University of Colorado at Boulder, Department of Physics
Physics 2020, Spring 2005
Lab 10
page 10 of 10
POTENTIAL EXAM QUESTIONS:
1. Paul sees a space-ship zooming by at 0.8c . Which of the following is/are true?
a) According to Paul’s measurements, the clocks on the ship are running slow.
b) Paul’s measurements of the ship’s clocks are incorrect because the ship is
moving at relativistic speed.
c) According to Paul’s measurements, the ship is shorter than it would be if it
slowed down.
d) Statements (a) and (c) are true.
e) Statements (a), (b), and (c) are true.
2. A 6 year old daughter named Earth-Child sees her hippie mother take off in a
rocket ship on her 30th birthday. The mother travels at 0.96c to a distant star and
returns to earth, having aged by 14 years, and arrives just as she is celebrating
her 44th birthday. How old is Earth-Child when her mom arrives home?
a) 10 years old
b) 20 years old
c) 56 years old
d) 76 years old
e) 185 years old
University of Colorado at Boulder, Department of Physics