Sunday, January 30, 2011

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?
 

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