Modern Physics

Kenneth S. Krane

Chapter 1

The Failures of Classical Physics - all with Video Answers

Educators

Chapter Questions

Problem 1

A hydrogen atom $\left(m=1.674 \times 10^{-27} \mathrm{~kg}\right)$ is moving with a velocity of $1.1250 \times 10^{7} \mathrm{~m} / \mathrm{s}$. It collides elastically with a helium atom $\left(m=6.646 \times 10^{-27} \mathrm{~kg}\right)$ at rest. After the collision, the hydrogen atom is found to be moving with a velocity of $-6.724 \times 10^{6} \mathrm{~m} / \mathrm{s}$ (in a direction opposite to its original motion). Find the velocity of the helium atom after the collision in two different ways: $(a)$ by applying conservation of momentum; $(b)$ by applying conservation of energy.

Brandy Heflin

Brandy Heflin

Numerade Educator

Problem 2

A helium atom $\left(m=6.6465 \times 10^{-27} \mathrm{~kg}\right)$ collides elastically with an oxygen atom $\left(m=2.6560 \times 10^{-26} \mathrm{~kg}\right)$ at rest. After the collision, the helium atom is observed to be moving with a velocity of $6.636 \times 10^{6} \mathrm{~m} / \mathrm{s}$ in a direction at an angle of $84.7^{\circ}$ relative to its original direction. The oxygen atom is observed to move at an angle of $-40.4^{\circ} .(a)$ Find the speed of the oxygen atom. ( $b$ ) Find the speed of the helium atom before the collision.

Matthew Baker

Numerade Educator

Problem 3

A beam of helium- 3 atoms $(m=3.016 \mathrm{u})$ is incident on a target of nitrogen- 14 atoms $(m=14.003 \mathrm{u})$ at rest. During the collision, a proton from the helium- 3 nucleus passes to the nitrogen nucleus, so that following the collision there are two atoms: an atom of "heavy hydrogen" (deuterium, $m=2.014 \mathrm{u}$ ) and an atom of oxygen- $15(m=15.003 \mathrm{u})$ The incident helium atoms are moving at a velocity of $6.346 \times 10^{6} \mathrm{~m} / \mathrm{s}$. After the collision, the deuterium atoms are observed to be moving forward (in the same direction as the initial helium atoms) with a velocity of $1.531 \times 10^{7} \mathrm{~m} / \mathrm{s}$.
(a) What is the final velocity of the oxygen- 15 atoms?
(b) Compare the total kinetic energies before and after the collision.

Suzanne W.

Numerade Educator

Problem 4

An atom of beryllium $(m=8.00 \mathrm{u})$ splits into two atoms of helium $(m=4.00 \mathrm{u})$ with the release of $92.2 \mathrm{keV}$ of energy. If the original beryllium atom is at rest, find the kinetic energies and speeds of the two helium atoms.

Suzanne W.

Numerade Educator

Problem 5

A 4.15-volt battery is connected across a parallel-plate capacitor. Illuminating the plates with ultraviolet light causes electrons to be emitted from the plates with a speed of $1.76 \times 10^{6} \mathrm{~m} / \mathrm{s} .$ (a) Suppose electrons are emitted near the center of the negative plate and travel perpendicular to that plate toward the opposite plate. Find the speed of the electrons when they reach the positive plate. $(b)$ Suppose instead that electrons are emitted perpendicular to the positive plate. Find their speed when they reach the negative plate.

Suzanne W.

Numerade Educator

Problem 6

Observer A, who is at rest in the laboratory, is studying a particle that is moving through the laboratory at a speed of $0.624 c$ and determines its lifetime to be $159 \mathrm{~ns}$. $(a)$ Observer A places markers in the laboratory at the locations where the particle is produced and where it decays. How far apart are those markers in the laboratory? (b) Observer B, who is traveling parallel to the particle at a speed of $0.624 c$, observes the particle to be at rest and measures its lifetime to be $124 \mathrm{~ns}$. According to $\mathrm{B}$, how far apart are the two markers in the laboratory?

Suzanne W.

Numerade Educator

Problem 7

A sample of argon gas is in a container at $35.0^{\circ} \mathrm{C}$ and 1.22 atm pressure. The radius of an argon atom (assumed spherical) is $0.710 \times 10^{-10} \mathrm{~m} .$ Calculate the fraction of the container volume actually occupied by the atoms.

Matthew Baker

Numerade Educator

Problem 8

By differentiating the expression for the MaxwellBoltzmann energy distribution, show that the peak of the distribution occurs at an energy of $\frac{1}{2} k T.$

Suzanne W.

Numerade Educator

Problem 9

A container holds $N$ molecules of nitrogen gas at $T=280 \mathrm{~K}$. Find the number of molecules with kinetic energies between $0.0300 \mathrm{eV}$ and $0.0312 \mathrm{eV}.$

Suzanne W.

Numerade Educator

Problem 10

A sample of 2.37 moles of an ideal diatomic gas experiences a temperature increase of $65.2 \mathrm{~K}$ at constant volume. (a) Find the increase in internal energy if only translational and rotational motions are possible. $(b)$ Find the increase in internal energy if translational, rotational, and vibrational motions are possible. $(c)$ How much of the energy calculated in $(a)$ and $(b)$ is translational kinetic energy?

Suzanne W.

Numerade Educator

Problem 11

An atom of mass $m_{1}=m$ moving in the $x$ direction with speed $v_{1}=v$ collides elastically with an atom of mass $m_{2}=3 m$ at rest. After the collision the first atom moves in the $y$ direction. Find the direction of motion of the second atom and the speeds of both atoms (in terms of $v$ ) after the collision.

Suzanne W.

Numerade Educator

Problem 12

An atom of mass $m_{1}=m$ moves in the positive $x$ direction with speed $v_{1}=v .$ It collides with and sticks to an atom of mass $m_{2}=2 m$ moving in the positive $y$ direction with speed $v_{2}=2 v / 3 .$ Find the resultant speed and direction of motion of the combination, and find the kinetic energy lost in this inelastic collision.

Suzanne W.

Numerade Educator

Problem 13

Suppose the beryllium atom of Problem 4 were not at rest, but instead moved in the positive $x$ direction and had a kinetic energy of $40.0 \mathrm{keV}$. One of the helium atoms is found to be moving in the positive $x$ direction. Find the direction of motion of the second helium, and find the velocity of each of the two helium atoms. Solve this problem in two different ways: $(a)$ by direct application of conservation of momentum and energy; $(b)$ by applying the results of Problem 4 to a frame of reference moving with the original beryllium atom and then switching to the reference frame in which the beryllium is moving.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 14

Suppose the beryllium atom of Problem 4 moves in the positive $x$ direction and has kinetic energy $60.0 \mathrm{keV}$. One helium atom is found to move at an angle of $30^{\circ}$ with respect to the $x$ axis. Find the direction of motion of the second helium atom and find the velocity of each helium atom. Work this problem in two ways as you did the previous problem. (Hint:
Consider one helium to be emitted with velocity components $v_{x}$ and $v_{v}$ in the beryllium rest frame. What is the relationship between $v_{x}$ and $v_{y}$ ? How do $v_{x}$ and $v_{y}$ change when we move in the $x$ direction at speed $v ?$ )

Paul A.

California State Polytechnic University, Pomona

Problem 15

A gas cylinder contains argon atoms $(m=40.0 \mathrm{u}) .$ The temperature is increased from $293 \mathrm{~K}\left(20^{\circ} \mathrm{C}\right)$ to $373 \mathrm{~K}\left(100^{\circ} \mathrm{C}\right)$. (a) What is the change in the average kinetic energy per atom? ( $b$ ) The container is resting on a table in the Earth's gravity. Find the change in the vertical position of the container that produces the same change in the average energy per atom found in part $(a)$.

Suzanne W.

Numerade Educator

Problem 16

Calculate the fraction of the molecules in a gas that are moving with translational kinetic energies between $0.02 k T$ and $0.04 k T.$

Suzanne W.

Numerade Educator

Problem 17

For a molecule of $\mathrm{O}_{2}$ at room temperature $(300 \mathrm{~K})$, calculate the average angular velocity for rotations about the $x^{\prime}$ or $y^{\prime}$ axes. The distance between the $\mathrm{O}$ atoms in the molecule is $0.121 \mathrm{nm}.$

Suzanne W.

Numerade Educator