Modern Physics: January 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

of the protons.

Express your answer as a fraction of the speed of light to three significant figures.

ANSWER:

0.6331

Part B
What is the kinetic energy of each proton?

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

ANSWER:

273.7

$\rm MeV$

Part C
What is the rest energy of the $\eta ^{0}$ ?

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

ANSWER:

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

must be done on a particle with a mass of

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:

1.399

Part B
How much work must be done on a particle with a mass of 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:

4.690

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:

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: ± Creating a Particle

Mastering Physics: Work Required to Accelerate Relativistic Particles

Mastering Physics: Relativistic Energy and Momentum

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