Modern Physics: 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

of a particle with mass

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

, in the laboratory frame of reference, of a proton moving with a speed of 0.899

. Use $938 \;{\rm MeV/c^2}$ for the mass of a proton.

Express your answer in $\rm MeV/c$ to three significant figures.

ANSWER:

1925

${\rm MeV/c}$

Part B

Find the total energy

of this proton in the laboratory frame.

Express your answer in millions of electron volts to three significant figures.

ANSWER:

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 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:

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?

Modern Physics

Sunday, January 30, 2011

Mastering Physics: Relativistic Energy and Momentum

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