Notes on Relativity and Cosmology - Physics Department, UCSB
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Notes on Relativity and Cosmology - Physics Department, UCSB

Notes on Relativity and Cosmology - Physics Department, UCSB Notes on Relativity and Cosmology - Physics Department, UCSB

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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.

6.7. HOMEWORK PROBLEMS 165 across a potential of one Volt 13 . Since the charge on an electron is 1.6 × 10 −19 Coulombs, one electron-Volt is 1.6 × 10 −19 J. (a) If an object at rest has a total energy of 1eV , what is it’s mass? (b) The mass of an electron is 9.11 × 10 −31 kg. What is the energy (in eV) of an electron at rest? (c) In a standard CRT (Cathode Ray Tube, like the tube in your TV or computer monitor), electrons are accelerated through a potential difference of about 5000 volts. In other words, moving through that potential adds 5000eV of energy to the electron. How fast is an electron going when it strikes your TV or computer screen? (d) Consider the electron that is just about to strike your screen. What is it’s momentum? If you used the old (and incorrect) formula p = mv to calculate its momentum, how much would the answer be off? 13 You can look at a PHY212 or 216 physics book for a definition of the Volt, but you won’t actually need to know it for this problem.

6.7. HOMEWORK PROBLEMS 165<br />

across a potential of <strong>on</strong>e Volt 13 . Since the charge <strong>on</strong> an electr<strong>on</strong> is 1.6 ×<br />

10 −19 Coulombs, <strong>on</strong>e electr<strong>on</strong>-Volt is 1.6 × 10 −19 J.<br />

(a) If an object at rest has a total energy of 1eV , what is it’s mass?<br />

(b) The mass of an electr<strong>on</strong> is 9.11 × 10 −31 kg. What is the energy (in<br />

eV) of an electr<strong>on</strong> at rest?<br />

(c) In a st<strong>and</strong>ard CRT (Cathode Ray Tube, like the tube in your TV<br />

or computer m<strong>on</strong>itor), electr<strong>on</strong>s are accelerated through a potential<br />

difference of about 5000 volts. In other words, moving through that<br />

potential adds 5000eV of energy to the electr<strong>on</strong>. How fast is an<br />

electr<strong>on</strong> going when it strikes your TV or computer screen?<br />

(d) C<strong>on</strong>sider the electr<strong>on</strong> that is just about to strike your screen. What is<br />

it’s momentum? If you used the old (<strong>and</strong> incorrect) formula p = mv<br />

to calculate its momentum, how much would the answer be off?<br />

13 You can look at a PHY212 or 216 physics book for a definiti<strong>on</strong> of the Volt, but you w<strong>on</strong>’t<br />

actually need to know it for this problem.

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