Physics 264L: Assignment 1
Made available: Monday, August 24, 2015
Due:
Monday, August 31, 2015, before class.
Problems
1. The Large Hadron Collider at CERN (European Organization for Nuclear Research) currently accelerates protons until they have a kinetic energy of about 7 TeV = 7 × 1012 eV. To two significant
digits
(a) What is the difference c − v between the speed of light and the speed of one of these protons?
Note: if you read the French text carefully, you can get this answer without needing a calculator,
by using an expression that was simplified using the fact that the rest mass of the proton energy
is small compared to its mass-energy in the LHC.
(b) What is the ratio of the mass of an LHC proton to the proton’s rest mass?
If someone had incorrectly thought that the 7 TeV was the proton’s total mass-energy, not its kinetic
energy, would that person get different answers to two significant digits?
2. The space shuttle with launch vehicle had a total mass of about 2 × 106 kg.
(a) Estimate to the nearest power of ten how high (in km) a grain of table salt (NaCl) could lift
this rocket, assuming perfect conversion of its mass energy m0 c2 to kinetic energy of the rocket.
Please state clearly all assumptions that you make to get your final estimate.
(b) Discuss briefly whether
i. it is possible in principle to extract the entire mass-energy of a salt grain. “In principle”
means that the laws of physics allow this, even though it might be difficult or expensive to
achieve in practice.
ii. if this is possible, is it possible in principle to convert all of that energy to the kinetic energy
of the rocket?
Some comments regarding 2(a): throughout this semester, I will be asking you to make order-ofmagnitude estimates to the nearest power-of-ten (no significant digits) problems, it is a useful and
important skill. For this particular problem
(a) first explain how you estimated the mass energy of a grain of table salt. Do not use the Internet
when getting your estimate (avoid being an Internet junkie who can’t do anything without looking
it up first), and don’t be overly precise, you only need to get a rough answer to the nearest power
of ten. It wouldn’t hurt for you to actually look at some grains of salt, e.g., spill some salt on the
table during a meal.
(b) show in detail how you got your final estimate without using a calculator, e.g., show what final
formula you obtained to represent the answer, explain how you approximated various numerical
quantities that entered into your formula, and show how you got your final estimate.
(c) the trick in doing order-of-ten estimates is to replace all numbers with the nearest power of ten
as soon as possible. With one exception (raising a number to an integer power), you should never
be multiplying or dividing decimal digits as you get your answer, and you will lose points in any
future exam question if you are asked to estimate an answer and you do multiply out digits. If
you have not practiced this estimation skill before, see this video:
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http://www.phy.duke.edu/~hsg/162/files/video-powers-of-ten-estimation.mp4
3. A negatively charged pion π − with rest mass mπ ≈ 140 MeV/c2 ≈ (1/7)mp is an unstable subatomic
particle that is frequently created in collisions of relativistic particles. After about 30 nanoseconds, a
pion at rest decays into a negatively charged muon µ− (this is an unstable fat electron, with rest mass
mµ ≈ 200me ) and a neutral neutrino ν which you can assume to have zero mass for this problem,
mν = 0. Use conservation of relativistic energy and momentum (Eq. (1-15) on page 21 of French and
Eq. (1-24) on page 24 of French) to show that the energy of the outgoing muon is given by
( 2
)
mπ + m2µ c2
Eµ =
.
(1)
2mπ
Is the muon moving relativistically, close to the speed of light?
4. Consider a point particle with rest mass m0 moving along the x-axis of some coordinate system under
the influence of a constant force of magnitude F , and assume the particle is initially at rest at the
origin. Classically (ignoring relativity), you know that the curve x(t) is a parabola satisfying x =
(1/2)(F/m0 )t2 . If you now treat the 1d motion using relativity, what kind of familiar mathematical
curve does x(t) trace out? Hint: pages 26-27 of the French text.
5. Problem 1-11 on page 31 of the French book.
6. Problem 1-12 on page 31 of the French book.
7. Problem 1-14 on page 32 of the French book.
8. Problem 1-19 on page 33 of the French book.
Note: if you are not familiar with using the generalized binomial theorem for simplifying powers of a
small quantity added to a larger quantity, look at Problem 1 of this Physics 162 assignment:
http://www.phy.duke.edu/~hsg/162/files/homeworks/assignment-01-jan-8-2015.pdf
9. Problem 2.4 on page 36 of the Taylor book on error analysis.
10. Problem 2.12 on page 38 of the Taylor book on error analysis.
11. To the nearest integer, please give the time in hours that it took you to complete this assignment,
including reading in the text.
Also, if you got help with this assignment, please give the names of the people from whom you got
help (classmates, the TA, myself, etc).
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12. Optional Challenge Problems
From time to time, I will include challenge homework problems that are substantially harder but
hopefully interesting and worthy for some of you. These challenge problems are completely optional,
do not feel obliged to try any of them. If you make some progress in solving any of these problems,
please write up what you have done and give it to me directly (not to the grader) and you can do this
at any time over the semester up to the last day of class. While I will not grade your answers nor give
you any official credit, any progress you make could have some positive influence on your final course
grade by the end of the semester.
I do ask though that you solve these on your own, without getting help from the Internet or from
people at Duke such as more senior students, graduate students or professors. Also, please do not work
on any of these challenge problems until after you have completed the assigned homework problems,
it is much more important that you master the basics first.
(a) Challenge problem 1.1: a simple mechanics problem that you learn to solve completely in an intro
physics course is the one-dimensional motion of a point particle of mass m as it is acted upon by
an ideal spring satisfying Hooke’s law, with force Fx = −mω 2 x, where ω is the angular frequency
of the simple harmonic motion. Solve the corresponding relativistic spring problem, in which
a point particle with rest mass m0 is acted on by the same simple force but now the particle’s
energy and momentum are in accord with special relativity. Does the particle still undergo simple
harmonic (sinusoidal) motion? If its maximum amplitude of oscillation is a, find an expression
for the period T of this relativistic oscillator in terms of ω and a, and show that
[
]
2π
3 ( ωa )2
T ≈
1+
+ ···
(2)
ω
16 c
for small amplitudes such that ωa/c ≪ 1 (one could call this the “weakly relativistic limit”).
(b) Challenge problem 1.2: a charged point particle moving in a uniform electic field traces out a
parabola. (This is just like projectile motion in a uniform gravitational field) . Consider now
an ultra-relativistic charged particle moving quite close to the speed of light with velocity v ≈
c cos(α)x̂ + c sin(α)ŷ with vy > 0 that enters a region of space filled with a uniform constant
electric field E = −E0 ŷ in the negative y-direction. Show that the path of the electron in this
electric field is a catenary.
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