PHYS 3310 Module 2 Self Assessment Practice Problems

Module 2 Self Assessment Practice Problems

2.1
Astronomers using a ground-based telescope are observing a distant star that is moving away from Earth at 0.555c. The star emits light at a frequency of 8.64 x 10¹⁴ Hz — a frequency that, in the laboratory, corresponds to a specific spectral line of hydrogen. What frequency do the astronomers measure?
Answer: 4.62 x 10¹⁴ Hz

2.2
A red dwarf star emits most of its light in the infrared and red portion of the electromagnetic spectrum. An observer at rest with respect to the star measures the peak wavelength of this emission to be 1000 nm. An astronomer using a ground-based telescope measures the same peak at 650 nm. Estimate the speed of the star with respect to Earth.
Answer: 0.403c

2.3
Muons are subatomic particles produced when cosmic rays collide with atoms in Earth’s upper atmosphere. They travel toward Earth’s surface at speeds close to the speed of light — fast enough that relativistic effects are necessary to describe their motion accurately. A muon traveling at 0.984c has a rest mass 207 times that of an electron.
(a) What is the muon’s relativistic momentum?
(b) What is the muon’s kinetic energy?
Answer: (a) 584 MeV/c (b) 488 MeV

2.4
Particle accelerators use large electric potential differences to accelerate charged particles to extreme speeds. In this problem, electrons are accelerated through a potential difference of 750 kV, giving each electron a kinetic energy of 7.5 x 10⁵ eV.
(a) Using relativistic mechanics, what is the ratio v/c for these electrons?
(b) What speed would classical mechanics predict for an electron with this kinetic energy? How does this compare to your answer in part (a), and what does the difference tell you?
Answer: (a) 0.914 (b) 1.71c

2.5
Protons are accelerated to extreme speeds in particle accelerators like the Large Hadron Collider at CERN, where understanding their relativistic properties is essential for predicting the outcomes of collisions. A proton is moving at 0.835c. Find its relativistic momentum, kinetic energy, and total energy.
Answer: 1423 MeV/c, 767 MeV, 1705 MeV

2.6
Two physicists, O and O′, are observing the same particle from different reference frames.
(a) Observer O measures the particle’s momentum to be 1256 MeV/c and its total relativistic energy to be 1351 MeV. What is the rest energy of this particle? Can you identify what particle it might be?
(b) Observer O′, moving in a different reference frame, measures the particle’s momentum to be 857 MeV/c. What total relativistic energy does O′ measure?
Answer: (a) 498 MeV (b) 991 MeV

2.7
In a particle accelerator, an electron and a proton are each accelerated from rest through a potential difference of 12.0 million volts. For each particle, find the relativistic momentum (in MeV/c) and kinetic energy (in MeV). Then calculate what classical mechanics would predict for each quantity and compare. What does the comparison tell you about when classical mechanics breaks down — and does it break down equally for both particles?
Answer: electron: 12.5 MeV/c and 12.0 MeV; proton: 150.5 MeV/c and 12.0 MeV

2.7

In a particle accelerator, an electron and a proton are each accelerated from rest through a potential difference of 12.0 million volts. For each particle, find the relativistic momentum (in MeV/c) and kinetic energy (in MeV). Then calculate what classical mechanics would predict for each quantity and compare. What does the comparison tell you about when classical mechanics breaks down — and does it break down equally for both particles?

Answer: electron: 12.5 MeV/c and 12.0 MeV; proton: 150.5 MeV/c and 12.0 MeV

2.8
Muons are unstable particles with a rest mass energy of 105.7 MeV. When a muon decays, it produces an electron and a neutrino — a nearly massless particle that carries away negligible energy in this approximation.
(a) Assuming all of the energy released in the decay goes into the electron’s kinetic energy, find the Lorentz factor γ for the resulting electron.
(b) What is the electron’s speed?
Answer: (a) 208 (b) 0.999988c

2.9
Nuclear fusion reactions power the sun and are the basis of ongoing efforts to develop fusion as a clean energy source on Earth. In both cases, light hydrogen isotopes fuse together and release energy in the form of gamma rays.
(a) Deuterium ( $^2$ H) captures a neutron to form tritium ( $^3$ H). Find the energy released as gamma radiation. Assume all kinetic energies are negligibly small. Atomic masses can be found in the table at the end of this module.
(b) Helium-3 ( $^3$ He) captures a neutron to form helium-4 ( $^4$ He). Find the energy released as gamma radiation under the same assumptions.
Answer: (a) 6.26 MeV (b) 20.8 MeV

2.10
Nuclear fission reactors generate electricity by splitting heavy uranium nuclei, releasing enormous amounts of energy in the process. In a typical reactor, each atom of uranium-235 releases about 210 MeV of energy when it fissions. If 1.50 kg of uranium-235 undergoes complete fission, what is the corresponding change in mass?
Answer: 1.4 grams