Problem Set 2
Due: see website for due dates
Chapter 2: Relativistic Mechanics
Question A
a. Athena asserts that a material particle must always have a speed less than that of
light because its mass continues to increase. Is she correct?
b. Next, Athena claims that even though photons have no mass, they can still travel at
the speed of light? Stated differently, can one prove that a massless particle must
always move at exactly the speed of light?
Question B
If you look up information on the proton, you’ll find that it’s (i) made of two up quarks and
a down quark and that the masses of the up quark has a mass of 2.4 MeV/c2, while the
down quark has a mass of 4.8 MeV/c2; so these sum for a total quark mass of 9.6
MeV/c2. However, the mass of a proton is 938 MeV/c2. Where does the rest of that mass
come from?
Question C
The amount of matter of an object does not depend on which frame you are in (i.e. mass is
a Lorentz invariant). Consider a head-on collision between two billiard balls (A-ball and Bball) where momentum and energy is transferred (assume the same rest masses, mA = mB).
a. From the reference frame of A, show that the B-ball "inherits" more mass-energy
than the A-ball.
b. From the reference frame of B, show that the A-ball "inherits" more mass-energy
than the B-ball.
c. Wrap-up this by showing that the "mass-energy transfer" between balls A and B
cannot depended on the choice of reference frame, therefore, mass is a Lorentz
invariant.
Question D
Photons are absorbed and reemitted as it travels through glass.
a. When photons go through glass, is there an increase in the mass of the glass?
b. It is a well-known fact that the light (large collection of “photons”) slows down while
going through glass, does this mean that the “photons” have gained mass?
Question E
a. Can a single photon spontaneous pair produce an electron-positron pair? Explain
your reasoning.
b. Can a single photon interacting with a nucleus pair produce an electron-positron
pair? Explain your reasoning.
Problem 1
(a) Compute the rest energy of 1-g of dirt. (b) If you could convert this energy entirely
into electrical energy (Hint: convert Joules into kWh) and sell it for 10 cents per kilowatthour, how much money would you get? (c) If you could power a 100-W light bulb with
the energy, for how long could you keep the bulb lit?
Problem 2
At the Stanford Linear Accelerator, electrons are accelerated to energies of 50 GeV. (a)
If this energy were classical kinetic energy, what would be the electrons’ speed? (b)
Calculate  and hence find the electrons’ actual speed.
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Problem 3
If an electron and proton (both initially at rest and far apart) come together to form a
hydrogen atom, 13.6 eV of energy is released (mostly as light). By how much does the
mass of an H atom differ from the sum of the electron and proton masses? What is the
fractional difference M/(me + mp)?
Problem 4
The K0 meson at rest decays into two charged pions, K0 →  + . Use conservation of
energy and momentum, to find the energies, momenta and speeds of the two pions.
(Give algebraic answer, in terms of the symbols mK and m first; then put in numbers.)
[Hint: mK = 498 MeV/c2 and m = 140 MeV/c2]
Problem 5
What is the speed of a particle that is observed to have momentum 500 MeV/c and
energy 1746 MeV? What is the particle’s mass (in MeV/c2)?
Problem 6
A neutral pion traveling along the x axis decays into two photons, one being ejected
exactly forward, the other exactly backward. The first photon has three times the energy
of the second. Prove that the original pion had speed 0.5c.
Problem 7
A particle of unknown mass M decays into two particles of known masses m1 = 0.5
GeV/c2 and m2 = 1 GeV/c2, whole momenta are measured to be p1 = 2.0 GeV/c along
the positive y axis and p2 = 1.5 GeV/c along the positive x axis. Find the unknown mass
M and its speed.
Problem 8
When a beam of high-energy protons collides with protons at rest in the laboratory (e.g.,
in a container of water or liquid hydrogen), neutral pions () are produced by the
reaction p + p  p + p + . Compute the threshold energy of the protons in the beam
for this reaction to occur.
Problem 9
The K0 and the 0 are elementary particles with masses mK and m, respectively. Find
the threshold energy for the reaction  + p  K0 +  0 in the frame of reference in which
the proton is at rest. Express your answer in terms of the masses of the particles.
Problem 10
a. A pion (0) has a lifetime of 8.4  1017s in its rest frame. In a high-energy collision, a
0 is produced with energy such that the relativistic gamma factor for the particle is
340. How far will the 0 travel before it decays?
b. Now suppose that the 0 decays into two massless particles. If one of the particles
moves in the same direction that the 0 was moving in, what is the direction of
motion of the second one?
c. What are the energies of each of the massless particles in terms of the momentum of
the original 0?
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