Homework Assignment 10
Physics 55
Made available:
Due in class:
Thursday, December 1, 2005
Monday, December 5, 2005
Problem 1: Problems Not in the Text
1. The energy-time uncertainty principle in quantum mechanics states that
∆E ∆t ≥
h
,
2π
(1)
where ∆t is the time spent observing some system, ∆E is the uncertainty in the total energy E of the
system, and h is Planck’s constant. This relation implies that if you observe a physical system for no
longer than an amount of time ∆t, then you can not measure the energy of the system to an accuracy
better than ∆E; long observation times are needed to determine precise energies.
Using Eq. (1), determine the largest value for the observation time ∆t such that the uncertainty in the
energy ∆E will exceed the rest mass energy 2me c2 of an electron-positron pair.
Indicate whether this amount of time is longer or shorter than the time that it takes light to cross the
diameter of a proton.
Note: Quantum mechanics allows the possibility that an electron-positron pair can actually materialize
out of nothing during this observation time provided the pair annihilates and disappears before this
interval of time is finished (and, crazily enough, this actually happens as shown by experiments).
2. Sam is five years older than his sister Sue. If Sam were to get into a relativistic rocket moving with
constant speed v = 0.8c, travel in a straight line away from the Earth, turn around instantly and return
with the same constant speed to Earth
(a) how far away from Earth (in light years and as measured by someone on Earth) would Sam have
to travel so that, upon his return to Earth, he will have reversed the age difference and end up
five years younger than his sister?
(b) how far did Sam measure himself to have traveled from Earth?
(c) how much older will his sister be when he returns?
3. The spacecraft Picard is moving directly away from the Earth with speed 0.9c when it launches a probe
containing a precise clock directly toward Earth. The probe has a proper length of 10 m and a proper
mass of 100 kg, and the probe’s clock emits radio pulses at precisely once a second in its rest frame. If
scientists on Earth measure the probe’s length to be 9 m as it travels toward Earth, what values will
scientists on the Picard measure for the speed, length, mass, energy, and time interval between radio
pulses of the probe?
4. At time t = 0 s, a point mass A at position x = 0 m on a ruler and an identical point mass B at
position x = 4 m on the same ruler start moving toward each other at a constant speed of 1 m/s
(A moves to the right, B moves to the left). At time t = 2 s, the masses collide and stick together for
two seconds as a motionless lump. The lump then disintegrates back into masses A and B, with A
moving to the left back toward the origin and with B moving to the right, both with speeds of 1 m/s.
1
Draw a clearly labeled spacetime diagram (see Figures S3.8 and S3.9 on pages 459-460 of the text)
with a horizontal space axis measured in units of meters and a vertical time axis measured in units of
seconds, and then draw on your spacetime diagram the worldlines of the two masses from time t = 0 s
to time t = 6 s.
5. Assume that identical indestructible high-precision atomic clocks are placed just above the surfaces of
(a) a black hole of one solar mass, (b) a neutron star of one solar mass, (c) the Sun, and (d) a white
dwarf of one solar mass. List these objects in order of slowest to fastest ticking of their clocks as viewed
by someone on Earth and justify your answer. (For this problem, you can assume none of these masses
is rotating.)
6. Consider a rigid metal circular ring of radius r that is rotating at a constant high speed about its
center. Explain why special relativity predicts that the familiar Euclidean formula C = 2πr (that
relates the circumference C of a circle to its radius) is not correct for this ring. Discuss whether the
rotating ring corresponds to the spherical or saddle-shaped geometries of Fig. S3.12 on page 462.
7. Consider two rockets that have the same length when they are resting on the ground and that pass
close by one other at a relative speed close to the speed of light as shown in Figure (a).
B'
B
bullet
B'
B'
O
O'
A'
A
O'
B
O'
O
O
A'
bullet
B
A'
O
A
A
(a)
(b)
(c)
The rocket on the right with observer O has a gun in its tail at point B that can shoot a bullet
perpendicular to the direction of relative motion. When observer O perceives the front of her rocket at
point A to just coincide with the rear of the other rocket at point A’, she fires the gun. Now according
to observer O, rocket O’ is highly contracted as shown in Figure (b) and so the bullet should clearly
miss rocket O’. (For the purpose of this problem, assume that the rockets are so close and are so narrow
in width that the time for the bullet to travel from one rocket to the position of the other is essentially
instantaneous.)
But now consider the situation from the point of view of observer O’, who is in the other rocket on
the left. He sees rocket O as greatly contracted and therefore sees the situation in Figure (c) when
points A and A’ coincide and so surely the bullet fired from O’s rocket will hit his rocket and cause
some damage.
2
(a) Identify where my description of this problem is not precise and so explain which panel, (b) or (c),
is not correct as drawn.
(b) Describe how the wrong panel should be drawn so as to be correct.
(c) Explain whether or not the bullet actually hits rocket O’. As a physical event at a particular point
in space (atoms actually get pushed around by the bullet, leaving a trace), everyone has to agree
whether the bullet struck or not, no matter who is the observer.
Problem 2: Extra Credit
1. Show algebraically that Einstein’s addition formula for two speeds can not produce a value greater
than the speed of light if the speeds themselves are less than or equal to the speed of light.
2. The most energetic particles observed in the universe are rare and extraordinary cosmic rays (protons)
that collide with the Earth’s atmosphere with an enormous kinetic energy of about 10 20 eV.
(a) How many protons would have to have their mass converted to energy (via E = mc 2 ) to provide
such a huge kinetic energy for one cosmic ray?
(b) According to an observer on Earth, such a cosmic ray would take about 100,000 years to zoom
across the Milky Way since such an energetic particle is traveling extremely close to the speed of
light and the Milky Way has a diameter of about 100,000 ly. But from the point of view of the
cosmic ray, the Milky Way is greatly contracted in length and so a much shorter time is needed
for the cosmic ray to cross the Milky Way.
i. According to the cosmic ray (more precisely, according to some observer moving with the
cosmic ray in the cosmic ray’s inertial frame of reference), what is the diameter of the Milky
Way?
ii. According to the cosmic ray, how long will it take to cross the Milky Way?
3. The following problem refers to the figure below. An enormous air table is set up in Duke’s soccer field
and painted with a coordinate system so that the x axis points toward the seats and the y axis points
toward one of the soccer goals. A large black straight stick is set in motion with a constant speed
in the y direction such that, exactly at 12 noon according to a Duke soccer fan sitting in the stands,
the entire stick coincides with the x axis as shown in part (a) below. At the exact moment that the
stick is just crossing the x-axis, the infamous Jackie zooms by from left to right along the x axis in her
relativistic rocket as shown, such that the center of her rocket just crosses the y-axis as the center of
the stick just crosses the x-axis.
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y
(a)
X
Rocket
y
y
(b)
X
(c)
X
(a) Using your knowledge of special relativity, explain why Jackie does not see a stick sitting on the
x axis as in part (a) of the figure but instead sees a tilted stick corresponding to either part (b)
or (c) in the above figure. Make sure to explain what she will actually see.
Hint: A standard trick is to try to map an unfamiliar problem to a familiar one. So assume in
the style of Einstein that the black stick is actually covered by many equally spaced flash bulbs
that are arranged to emit a bright flash when the part of the stick just below a given flash bulb
crosses the x axis. Then the Duke soccer fan will see all the lights flash at the same time when
the stick just crosses the x axis as in part (a). But as Jackie zooms by, what does she see in terms
of the lights on the stick going off?
(b) Calculate the value of the angle θ = θ(v) that the Jackie observes between the stick and x-axis.
Problem 3: Comments about the Homework and Course
• About how long did it take you to complete this assignment?
• Do you feel that you are understanding the course material? If not, please indicate what topics or
ideas you would like to understand better.
• Comments or suggestions about other parts of the course such as reading, homeworks, lectures, or
observation sessions?
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