Due 5 p.m., Friday, November 20
Revised, November 17:
Problem 3: minor wording change.
Problems 4 and 5: hints added.
Problem 6: removed from the assignment.
Problem 9: note expanded.
Revised, November 20:
Problem 5: minor typos corrected
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Reading and topic coverage
Throughout this assignment, “R” refers to Six Ideas That Shaped Physics: Unit R:
The Laws of Physics Are Frame-Independent , Second Edition, by Thomas Moore.
Here is the tentative plan for upcoming classes. As always, this is subject to change.
Fri., Nov. 12 § R10 Conservation of four-momentum
Creating massive particles
Mon., Nov. 15 § R10.4
Wed., Nov. 17 § R10
Momentum and energy of photons
Dr. William Phillips visit
Finish up conservation of four-momentum
Fri., Nov. 19 Handouts Begin cosmology
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Problems to work but not turn in.
A. The Large Hadron Collider (LHC) is a particle accelerator under development at CERN, just outside Geneva, Switzerland. It will soon be able to accelerate protons to a total energy of 7 TeV.
(a) The mass of a proton is 1 .
672 × 10
− 27 kg. What is its rest-mass-energy in
J? In eV? (Look up the conversion from Joules to eV if you don’t have it handy.)
(b) What is the Lorentz factor, γ of the protons in the LHC?
(c) What is the velocity of the protons in the LHC?
B. Problem R10B.5.
C. Problem R10B.6.
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Problems to turn in.
1. Problem R10B.3.
Continued on the next page....

2. Problem R10T.3 parts a through e.
Notes.
(i) All electrons have the same mass. (ii) In part (d), all the light energy is emitted in the same direction (“out the back of the ship”).
3.
(a) Answer the same question as R10T.3 part c, except that after the collision, two particles less massive than electrons move north and south at smaller speeds than the original electron speeds. Could this be consistent with conservation of four-momentum for some such speed?
(b) Answer the same question as R10T.3 part c, except that after the collision, two particles more massive than electrons move north and south at smaller speeds than the original electron speeds. Could this be consistent with conservation of four-momentum for some such speed?
(c) In R10T.3 part (d), how could the emission of the light-energy from the ship be changed (compared to the description given in the previous problem) in order to switch your original answer (from “consistent” to
“false/inconsistent” or from “false/inconsistent” to “consistent”)?
4. A particle of mass M
0 is at rest when it decays into three identical particles, each of rest mass m
0
. Two of the particles have velocities and directions as shown below.
(a) Calculate the direction and speed of particle 3.
(b) Find the ratio M
0
/m
0
.
v
3
=?
v
1
=
4
5 v
2
=
3
5
Hint.
This is grungy. Suggested procedure: (i) Write out the conservation of four-momentum, putting in all known numerical values and writing the unknown four-momentum of particle 3 in terms of unknown quantities γ
3
, v
3 x
, and v
3 y
. (ii) Write out the conservation rules you get from each of the individual components of four-momentum. You should be able to find expressions for
M
0
/m
0
, v
3 x
, and v
3 y
; each expression will depend on the as-yet-unknown value of γ
3
. (iii) Use trigonometry to find θ . (iv) The Pythagorean theorem tells you that v
2
3
= v
2
3 x
+ v
2
3 y
. Plug your expressions for v
3 x and v
3 y into this. Then note that
γ
3
= √ 1
1 − v
2
3
⇒ γ 2
3
= √ 1
1 − v
2
3
2
=
1
1 − v
2
3
⇒ 1
γ
2
3
= 1 − v 2
3
.
Substitute this in to your expression for v 2
3
, which should leave you with an equation in which the only unknown is and then M
0 m
0
.
v
3
. Solve for v
3
. (iv) Use v
3 to find γ
3
Continued on the next page....

5. Problem R10S.8.
Hint.
Another grungy problem. There are different ways to approach it. Here’s one method. I’ll use the Greek letter nu, ν , to denote the neutrino and the Greek letter mu, µ , to denote the muon. Write out conservation of four-momentum, leaving the neutrino energy, E
ν
, the muon energy, E
µ
, and the muon momentum, p
µ as unknowns. (Note that, because the neutrino is nearly massless, you can assume that p
ν
= E
ν
, so the momentum four-vector for the neutrino can be written in terms of E
ν alone.) This gives you two equations relating three unknowns ( E
ν
, E
µ
, and p
µ
). We know that, in general, m 2 = E 2 − p 2 . For the muon, you are given its mass, m
µ
(106 MeV)
2
= E
2
µ
− p
2
µ
= 106 MeV, so you can write that
. This gives you a third equation relating the three unknowns. Algebraic manipulation of the three equations will allow you to solve for E
µ
, from which you can find the Lorentz factor of the muon and hence its speed.
6. Problem 6 has been removed from this assignment.
(It was formerly problem R10S.9. The solution to this problem will be given in the homework solution.)
7. Problem R10S.6. Instead of using the value of m = 25 t given in the problem, write your answer for M as a multiple of m . (In fact, the answer to this problem is given in the beginning of problem R10R.1.)
8. Problem R10R.1.
Note.
Our Galaxy is a around 60,000 years across. A light-year is the same thing we have been calling a “year” in special relativity.
Continued on the next page....

9. An article from the November 10 New York times describing LightSail-1 was distributed in class last week (web link given below). This is an experimental spacecraft with a 32 m 2 mylar “sail” which will intercept sunlight and use the momentum of the sunlight to propel the spacecraft. The mass of LightSail-1 is around 5 kg. The intensity of sunlight near the Earth is 1400 W/m
2
.
Assume that LightSail-1 starts at rest and is oriented so that sunlight falls perpendicular onto its sail. Assume all the sunlight momentum falling onto it is converted into momentum of the spacecraft. How fast will it be going after one day?
Note.
This is mostly a problem in unit conversion. Find the energy intercepted by the sail in Joules and convert it to special relativity units. In special relativity units, E = p , so this immediately gives you momentum in special relativity units. Convert the momentum to units of m kg m and then find the velocity in s using p = mv . The speed is small so it is OK to use the non-relativistic s momentum formula here.
The New York Times article is here: http://www.nytimes.com/2009/11/10/science/space/10solar.html
You can read more about it at the Planetary Society website: http://www.planetary.org/programs/projects/solar sailing