Sunday, January 30, 2011

Mastering Physics: ± Creating a Particle

Two protons (each with rest mass m =1.67 \times 10^{-27}\;{\rm kg}) are initially moving with equal speeds in opposite directions. The protons continue to exist after a collision that produces an \eta^{0} particle. The rest mass of the \eta^{0} is m_\eta=9.75 \times 10^{-28}\;{\rm kg}.

 
Part A
If the two protons and the \eta^{0} are all at rest after the collision, find the initial speed v of the protons.
Express your answer as a fraction of the speed of light to three significant figures.
ANSWER:

  v  = 0.6331
  c
 
Part B
What is the kinetic energy E_k of each proton?
Express your answer in millions of electron volts to three significant figures.
ANSWER:

  E_k  = 273.7
  \rm MeV
 
Part C
What is the rest energy E_r of the \eta ^{0}?
Express your answer in millions of electron volts to three significant figures.
ANSWER:

  E_r  = 547.3
  \rm MeV
Notice that the rest energy of the \eta^{0} is equal to the total kinetic energy of the two protons, as expected from the conservation of energy.
 

Mastering Physics: Work Required to Accelerate Relativistic Particles

Part A
How much work W must be done on a particle with a mass of m to accelerate it from rest to a speed of 0.909 {\it c}?
Express your answer as a multiple of mc^{2} to three significant figures.
ANSWER:

  W  = 1.399
  mc^2
Part B
How much work W must be done on a particle with a mass of m to accelerate it from a speed of 0.909 {\it c} to a speed of 0.990 {\it c}?
Express your answer as a multiple of mc^{2} to three significant figures.
ANSWER:

  W  = 4.690
  mc^2
 

Mastering Physics: Relativistic Energy and Momentum

Learning Goal: To learn to calculate energy and momentum for relativistic particles and, from the relativistic equations, to find relations between a particle's energy and its momentum through its mass.
The relativistic momentum p_vec and energy E of a particle with mass m moving with velocity v_vec are given by
\vec{p} = \frac{m\vec{v}}{\sqrt{1-\frac{v^2}{c^2}}}
and
E = \frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}}.
Part A
Find the momentum p, in the laboratory frame of reference, of a proton moving with a speed of 0.899 c. Use 938 \;{\rm MeV/c^2} for the mass of a proton.
Express your answer in \rm MeV/c to three significant figures.
ANSWER:

  p = 1925
 {\rm MeV/c}

Part B
Find the total energy E of this proton in the laboratory frame.
Express your answer in millions of electron volts to three significant figures.
ANSWER:

  E = 2142
 {\rm MeV}

Part C
What is the value of the expression E^2-(pc)^2 for this proton?
Express your answer in millions of electron volts squared to three significant figures.
ANSWER:

  E^2-(pc)^2 = 8.80×105
 {\rm MeV^2}
The answer to this part is numerically equal to the rest mass of the proton squared. In fact, this result points to a very important equation in relativistic physics: E^2-(pc)^2 = m^2c^4. This formula indicates an equivalence between mass and energy, which was first realized by Albert Einstein.
 
Part D
What is the rest mass m of a particle traveling with the speed of light in the laboratory frame.
Express your answer in {\rm MeV/c^2} to one decimal place.
ANSWER:

  m = 0.0
 {\rm MeV/c^2}
Thus, if a particle moves at the speed of light, it must have no rest mass. Usually this fact is stated somewhat differently: Massless particles always move at the speed of light. Can you think of an example of such particles?