Flat Prairie Publishing

Marian Physics

Special Relativity

1 SpaceTimeSpecial Relativity

Space & Time

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Problem Set RP 1 Special relativitySection 1: Length contraction, time dilation, velocity addition, transformation equations.

1. The proper mean lifetime of pions is 2.6 * 10 -8 s.  A beam of pions has a speed of 0.85c with respect to the laboratory.

a) What will be the mean lifetime of the pions as measured by a clock in the laboratory?

b) How far will they travel on average, before they decay?

c)  What would be your answer to part b if you neglected spe-cial relativity?

2.  A muon has an average lifetime (proper mean lifetime) of 2.2 * 10-6 s.  Muons in a particle beam are travelling at 0.999c with respect to the laboratory.

a) What is their mean lifetime as measured in the laboratory frame of reference?

b) How far will the muon travel in the laboratory frame before it decays?

Section 1

Space and TimeSpecial Relativity

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3.  Francie concocts a brew of fast moving pions.  She meas-ures their mean lifetime to be 7.5 *10 -8 s.  She knows their proper lifetime when not moving is 2.6 * 10 -8 s.  How fast are the pions moving with respect to Francie?

4.  Cosmic rays from space collide with the nuclei of atoms in the Earth’s upper atmosphere, producing elementary parti-cles called muons.  Muons are unstable and decay after an average lifetime of 2.2 * 10 -6 s.  That is the number of muons in a given sample should decrease in time according to:

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Where N is the number of remaining muons after time t, N0 is the number of muons at time t=0 and is the mean lifetime of the particle.

In an experiment at the top of Mt. Washington in New Hamp-shire a detector counted 563 muons/hr moving downward at a speed of 0.9952c.  At sea level, 1907 m below the first measure-ment the detector counted 408 muons/hr. (Frish, David H. and Smith James H. “Measurement of the Relativistic Time Dilation Using Mesons” American Journal of Physics, 31, 342, 1963)

a) According to Galilean relativity , what would you expect the counter at sea level to be?

b) According to special relativity, what would you expect the counter at sea level to be?

5.  Megan is in charge of the starship “Frostboss” which moves away from Earth with a speed of 0.8c.  Jess takes the starship “Hood” and moves away from Earth in the opposite direction  at a speed of 0.6c with respect to Earth.  What is the speed of Frostboss as measured by Hood.

6.  Amanda has extraordinary strength.  She takes a rod of length L and throws it toward Kate. As the rod passes Kate, Kate measures the length of the rod to be L/2.  What is the speed of the rod with respect to Kate?

7. Ned is sitting on the North Bend train station waiting for Melissa. Earlier, he had measured the platform to be 60 m long. A high speed Union Pacific train approaches, Melissa at the con-trols.  As the train goes by he notices that the front and back ends of the train line up exactly with the ends of the platform at the same time. Melissa notices that her speedometer reads 0.6c (fast train) with respect to the ground.  Ned waves to Melissa as the train goes by.  What is the length of the train ac-cording to Melissa?

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8.  Amanda throws a meter stick with a velocity of 0.8c with re-spect to you in a direction parallel to the length of the stick.  Find the length of the stick as measured by you.

9.  Molly is excited as she learns that she will be piloting the first starship to travel to Alpha Centauri.  Ned is not so sure as he is certain that Molly will not be back in time for senior prom.  Molly explains to Ned that Alpha Centauri is only 4 light years away from Earth.  This makes Ned feel better as 4 is a small number.  Molly travels to Alpha Centauri at a speed of 0.75 c with respect to Earth.

a)  How long does the trip take to Alpha Centauri as measured by a clock on Earth?

b)  How long does the trip take as measured by Molly’s watch?

c) What is the distance between Earth and Alpha Centauri as measured by Molly?

10.  Melissa and Kathy discover that they were born on the same day.  Melissa being the adventurous type, takes a cruise to Frobozz, a small planet orbiting a star 15 light years away from Earth. Frobozz is known for its fun times and sensuous … well I digress … The cruise ship travels to Frobozz at 0.999c, Melissa stays on the planet for 10 years then returns to earth on the same ship, travelling at the same speed.

a) How long has Melissa been gone as measured by Kathy?

b) How long has Melissa been gone as measured by Melissa?

c) How much older is Kathy than Melissa is when Melissa reaches Earth?

11. (hard and optional)

A friend of yours who is the same age as you travels to Alpha Centauri (4 light years away from Earth) and back.  She claims that the trip only took 6 years. How fast did she travel?

Section 2: Rest Energy, Momentum, Kinetic Energy, Relativ-istic Energy and Mass.

12. How much mass must be converted to energy to produce on Joule of energy?

13.  Sketch a graph of the momentum of a particle vs its speed V.  For v units should be as a fraction of c. (i.e. 0.1c, 0.2c, 0.3c etc)

14.  Da watt is a unit of power... a measure of the use/consumption/generation/creation/transfer of energy .  One watt is one joule per second.

a) Calculate the rest energy of one gram of dirt in Joules?

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b)  If you could convert this energy into electrical energy at 90% efficiency, how much electrical energy could you obtain?

c) If you could sell this for 10 cents per kilowatt-hour, how much money would you get?

d) If you powered a 100 watt light bulb with this energy how long would the bulb stay lit?

15. An electron with rest energy of 0.511MeV moves with speed 0.2c.  Find its total energy, kinetic energy and momentum.  

16. A free neutron decays into a proton and electron.

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a) How much energy is released in this reaction?

b) In order to generate 1,000 watts, how many neutrons would have to decay each second?

17.  How much energy would be required to accelerate a parti-cle of mass m from rest to

a) 0.5c     b) 0.9c  c) 0.99c

Express your answer in multiples of the rest energy.

18.  A muon has a rest energy of 105.7 MeV. Calculate its rest mass in Kilograms.

19.  A proton with rest energy of 938 MeV has a total energy of 1,400 MeV

a)  What is its speed?

b)  What is its momentum?

20. In a typical nuclear fusion reaction a tritium nucleus and

a deuterium nucleus fuse together to form a helium nucleus plus a neutron.  How much energy is released in this fusion re-action?

21.  A deuteron consists of a proton and neutron bound to-gether (heavy hydrogen).  It is the nucleus of the deuterium atom, which is an isotope of Hydrogen, called heavy hydrogen

and is written .  How much energy is required to separate the proton from the neutron in the deuteron?

The energy needed to break up a nucleus into its constituent parts is called the binding energy of the nucleus.  As you calcu-lated (I hope), the binding energy of the deuteron is 2.22MeV. This is the energy that has to be added to the nucleus to break it up into a proton and a neutron.  This can be done by bombard-ing deuterons with energetic particles or electromagnetic radia-tion with energy of at least 2.22 MeV.  

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When a deuteron is formed by the combination of a neutron and a proton energy will be released. When neutrons from a re-actor collide with protons, some neutrons are captured to form deuterons. Although such capture is officially considered to be a violation of the deuteron capture act passed by congress in 1967 it still occurs today. Such a capture is considered to be a violation of the deuterons civil rights and certain legal types are considering a class action lawsuit … but I digress  … It should be obvious  that when a proton captures a neutron 2.22 MeV of energy is released, usually in the form of electromagnetic radia-tion.

22. The sun radiates energy at the rate of 4 * 1026 watts. As-sume that this energy is produced by the reaction whose net re-sult is the fusion of 4 hydrogen nuclei to form 1 He nucleus (this is a gross simplification but will suffice for this approximation ex-ercise).  

a) How much energy is released for each He nucleus formed?

b) Calculate the sun’s loss of rest mass each day.  

Assuming (actually another bad assumption but illustrative of a first approximation technique and upper limit to the issue) that the entire mass of the sun will be consumed by this type of reac-tion,

c) how long will there be a sun?

Section 3 Red Shift:

23. How fast must you be travelling towards a red light whose wavelength is 650 nm for it to appear green (wavelength 525 nm)?

24. Sodium light of wavelength 589 nm is emitted by a source that is receeding from the Earth with a speed V.  The wave-length measured by an observer on Earth is 620 nm.  Find V

25. In its rest frame the quasar PC 1247+3406 produces a hy-drogen emission line of wavelength 1216 angstroms. On Earth, this emission line is observed to have a wavelength of 7214 angstroms.

a) Find the redshift parameter for this quasar.

b) Calculate the speed of recession for this quasar.

(data from Schneider et al (1991).  As of July 1998 this quasar has the greatest redshift yet measured.  Shneider, Donald P. Schmidt, Maarten and Gunn James E. “PC 1247-3406: An Opti-cally Selected Quasar with a Redshift of 4.897”  The Astronomi-cal Journal, 102, 837, 1991

26. In its rest frame quasar Q2203+29 produces a hydrogen emission line of wavelength 1216 angstroms. Astronomers on Earth measure a wavelength of 6568 angstroms for this line. De-termine the redshift parameter and speed of this quasar.

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  McCarthy, Patrick J. et al. “Serendipitous Discovery of a Red-shift 4.4 QSO” The Astrophysical Journal Letters 328, L29, 1988.

27 Quasars are thought to be the nuclei of active galaxies in the early stages of their formation.  A typical quasar radiates energy at the rate of 10^41 watts.  At what rate is the mass of this qua-sar being reduced to supply this energy?  Express your answer in solar mass units per year.