Notes on Relativity and Cosmology - Physics Department, UCSB

164 CHAPTER 6. DYNAMICS: ENERGY AND ...
In relativity, mass and energy are not conserved separately. Mass and energy
in some sense merge into a single concept ‘mass-energy 12 .’ Also, we have seen
that energy and momentum fit together into a single spacetime vector just
as space and time displacements fit together into a ‘spacetime displacement’
vector. Thus, the concepts of momentum and energy also merge into a single
‘energy-momentum vector.’
6.7 Homework Problems
Note: Energy can be measured in various units, like Joules (J) or kiloWatthours
(kW-hrs., this is the unit that Niagara Mohawk uses on my electric bill).
You can use any unit that you like. You may find the following relations between
the various units useful.
1kg(m 2 /s 2 ) = 1Joule(J) = 1Watt − second(Ws) = 1
3.6 × 10−6 kW − hrs.
6-1. How much energy would take to accelerate you up to .9c?
6-2. I pay Niagara Mohawk $0.107 per kW-hr. How much would it cost me to
accelerate you up to .9c?
6-3. Consider a box containing two photons traveling in opposite directions.
If the box has a rest mass m 0 and each photon has an energy E 0 , what
is the rest mass of the combined box-plus-photons system? Hint: How
much energy and momentum does each of the three objects have?
6-4. In particle accelerators, one can collide an electron with a positron and
(sometimes) they turn into a proton/anti-proton pair. The rest mass of an
electron (or a positron) is 9.11 × 10 −31 kg. The rest mass of a proton (or
an anti-proton) is 1.673 × 10 −27 kg. Suppose that the proton/anti-proton
pair is created at rest and that the electron and positron had equal speed
in opposite directions. How fast must the electron and positron have been
moving for this reaction to be allowed by conservation of energy? Give the
answer both in terms of speed v and boost parameter θ.
6-5. Here’s a good calculation if you know a little physics. It has to do with
how your TV and computer monitor work:
Particle physicists often use a unit of energy called the “electron-Volt”
(eV). This amount of energy that an electron picks up when it accelerates
12 Usually just called ‘energy’ in modern terminology.