Competitive Physics: Thermodynamics, Electromagnetism and Relativity

Wang Jinhui

Chapter 12 Relativistic Dynamics - all with Video Answers

Educators

Chapter Questions

Problem 1

Tom proposes the following mechanism to generate "infinite" energy. Orient two perfectly reflective mirrors (of arbitrary masses) such that they are mutually parallel and stationary initially. Now, place a photon between the two mirrors such that it impinges the mirrors normally. As the photon bounces back and forth between the two mirrors, it imparts momentum and thus kinetic energy to the two mirrors. Furthermore, the photon can always catch up with the mirrors, even if the mirrors begin to pick up speed so this process will continue indefinitely - producing "infinite" kinetic energy. What is wrong with Tom's reasoning? Now, consider a new set-up where the two mirrors are identical and initially stationary. Two photons, with initial velocities in opposite directions and initial frequency $f$ each, impinge normally on the two mirrors repeatedly. What is the final kinetic energy of each mirror after a long time?

Ren Jie Tuieng

Numerade Educator

03:00

Problem 2

A particle of rest mass $m$ is traveling in the positive $\mathrm{x}$-direction at velocity $u$ in the lab frame. It then disintegrates into two identical particles of rest mass $\frac{m}{\sqrt{6}}$ each. Determine the velocities of the product particles in the lab frame if they are aligned with the $\mathrm{x}$-axis.

Julie Farhm

Numerade Educator

01:35

Problem 3

A particle of rest mass $m_1$ is bombarded at another stationary particle of rest mass $m_2$ at initial velocity $u$. If this collision triggers the production of a third particle (while retaining the rest masses of the other two), determine the maximum rest mass of the third particle.

Penny Riley

Numerade Educator

02:26

Problem 4

A particle of unknown mass $M$ decays into two particles of known masses $m_a=0.5 \mathrm{GeV} / c^2$ and $m_b=1.0 \mathrm{GeV} / c^2$, whose momenta are measured to be $p_a=2.0 \mathrm{GeV} / c$ directed along the $\mathrm{y}$-axis and $p_b=1.5 \mathrm{GeV} / c$ directed along the x-axis. Find the unknown mass $M$ and its speed (in units of $c$ ).

Chai Santi

Numerade Educator

05:03

Problem 5

Consider the reaction
$$
e^{+}+e^{-} \rightarrow e^{+}+e^{-}+\psi .
$$
Determine the minimum initial energy of the electron or positron for this reaction to occur in the center-of-momentum $(\mathrm{CoM})$ frame in terms of the rest masses $m_\psi$ and $m_e$. Hence, find the threshold energy of the positron if the positron is bombarded at the stationary electron in the lab frame, without the aid of Eq. (12.26).

Sam Stansfield

Numerade Educator

01:05

Problem 6

A rocket is initially stationary with a rest mass $M_i$ in the lab frame $\mathrm{S}$. The rocket then begins to convert mass into photons and ejects them from the back. When the rest mass of the rocket is $M_f$, prove that its speed $u$ in frame S fulfils
$$
\frac{M_i}{M_f}=\left(\frac{1+u}{1-u}\right)^{\frac{1}{2}} .
$$

Raj Bala

Numerade Educator

02:27

Problem 7

In the lab frame $\mathrm{S}$, a relativistic rocket of initial rest mass $M_i$ is initially stationary. The rocket then begins to eject mass continuously in minuscule amounts at one instant, towards the back, at a velocity $u$ relative to its instantaneous rest frame. When the rest mass of the rocket is $M_f$, prove that the velocity $v$ of the rocket in frame $\mathrm{S}$ satisfies
$$
\frac{M_i}{M_f}=\left(\frac{1+v}{1-v}\right)^{\frac{1}{2 u}}
$$

Khoobchandra Agrawal

Numerade Educator

06:34

Problem 8

A bucket of initial rest mass $M_0$ has an initial velocity $u_0$ in frame $S$. It begins to collect sand aligned in a line with a linear mass density $\lambda$ in S. Assuming that the line of sand extends forever,
(a) Find the rate of rest mass increase of the bucket when the bucket has speed $u$. Why is this greater than $\lambda u$ ? Find the rest mass of the bucket, $M(t)$, as a function of time.
(b) Find the energy and velocity of the bucket as functions of time, $E(t)$ and $u(t)$.
(c) Find the energy and velocity of the bucket as functions of the displacement of the bucket relative to the bucket's initial position, $E(x)$ and $u(x)$.
(Adapted from "Introduction to Mechanics")

Katie Mcalpine

Numerade Educator

15:30

Problem 9

Referring to the previous scenario, the bucket-and-contained-sand system now loses a fraction $f$ of its remaining rest mass per unit distance traveled. Find $E(x), p(x)$ and $t(x)$. (Adapted from "Introduction to Mechanics")
Problems with Four-Vectors
A common trick in solving equations involving four-momenta involves isolating a single four-momentum that is not of interest and then taking the squared norm of both sides to eliminate the irrelevant four-momentum (as its squared norm produces the mass of the particle). This will be a common denominator in many problems.

Hafiz Shahzaib

Numerade Educator

Problem 10

(a) Let $A$ be a four-vector, and suppose that one component of $A$ is found to be zero in all inertial frames. Show that all four components of $A$ are zero in all frames. This is known as the zero-component theorem.
(b) The four-momentum of a particle in the lab frame $\mathrm{S}$ is $P$ while the fourvelocity of an observer is $U$ with respect to $\mathrm{S}$. Show that the particle's energy in the rest frame of the observer is $P \cdot U$.
(c) Prove that the inner product of the four-velocity $U$ and the fouracceleration $A$ of a massive particle is zero.
(d) Show that the instantaneous charge density $\rho$ and the instantaneous current density $\boldsymbol{j}$ at a particular location forms a four-vector $J=(\rho c, \boldsymbol{j})$ in an arbitrary inertial frame $\mathrm{S} . J$ is known as the four-current. Hint: consider the four-velocity.

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02:58

Problem 11

A particle $m_a$ with speed $v_a$ is pursuing another particle $m_b$ with $v_b\left(v_b<v_a\right)$ along the $\mathrm{x}$-axis of an inertial frame $\mathrm{S}$. When particle a catches up with particle $\mathrm{b}$, they collide and coalesce to form a single particle of mass $m$. Show that
$$
m^2=m_a^2+m_b^2+2 m_a m_b \gamma_{v_a} \gamma_{v_b}\left(1-\frac{v_a v_b}{c^2}\right) .
$$

Dheeraj Sharma

Numerade Educator

03:07

Problem 12

A particle of rest mass $m_1$ is initially stationary in the lab frame $\mathrm{S}$. It then disintegrates into a photon and another particle of rest mass $m_2<m_1$. Find the energies of the photon and the final particle.

Suzanne W.

Numerade Educator

03:37

Problem 13

In an inertial frame $\mathrm{S}$, a photon of frequency $f$ is normally incident on a perfect plane mirror, of mass $m$, retracting at a velocity $u$ from the photon. Find the frequency $f^{\prime}$ of the reflected photon in frame $\mathrm{S}$.

Nick Johnson

Numerade Educator

01:55

Problem 14

A proton of energy $E$ collides elastically with a second proton of rest energy $E_0$ that is initially stationary. Subsequently, the two protons are directed at angles $\pm \frac{\phi}{2}$ relative to the initial velocity of the incident proton. Find $\cos \phi$.

Mayukh Banik

Numerade Educator

08:08

Problem 15

A massive particle with energy $E_0$ and speed $\beta_0 c$ undergoes a head-on elastic collision with a photon with energy $E_{\gamma 0}$. Show that the final energy $E_\gamma$ of the photon is
$$
E_\gamma=E_0 \frac{1+\beta_0}{2+\left(1-\beta_0\right) \frac{E_0}{E_{\gamma 0}}} .
$$
Show that $E_\gamma<E_0$ but if $\beta_0 \rightarrow 1, \frac{E_\gamma}{E_0} \rightarrow 1$. That is, a high-energy particle loses most of its energy to the photon (in the ideal $\beta_0 \rightarrow 1$ limit, the particle retains energy $E_{\gamma 0}$ by the conservation of energy).

David Morabito

Numerade Educator

04:47

Problem 16

16. Emission by Excited Atom*
An excited atom $A^*$ at rest drops to its ground state $A$ by emitting a photon. In atomic physics, it is usually assumed that the energy $E_\gamma$ of the emitted photon is equal to the difference in energies of the two atomic states, $\Delta E=\left(M^*-M\right) c^2$, where $M$ and $M^*$ are the rest masses of the ground and excited states of the atom. This cannot be exactly true, since the recoiling atom $\mathrm{X}$ must carry away part of $\Delta E$. Show that in fact
$$
E_\gamma=\Delta E\left(1-\frac{\Delta E}{2 M^* c^2}\right) .
$$
Given that $\Delta E$ is of order $\mathrm{eV}$ while the lightest atom has $M$ of order $\mathrm{GeV} / c^2$. Discuss the validity of the approximation $E_\gamma=\Delta E$.

João Bravo

Numerade Educator

04:31

Problem 17

High-energy neutrino beams are produced by allowing an excited pion $\pi^{+}$ to decay into an excited muon $\mu^{+}$and neutrino $\nu$ according to the equation
$$
\pi^{+} \rightarrow \mu^{+}+\nu
$$
The masses of a pion and muon are $140 \mathrm{MeV} / c^2$ and $106 \mathrm{MeV} / c^2$. The mass of a neutrino is negligible.
(a) Find the energy of the neutrino in the rest frame of the pion.
(b) In the lab frame, the pion has an energy of $200 \mathrm{GeV}$. If the momentum of the neutrino is aligned with the initial momentum of the pion, determine the energy of the neutrino.
(c) Referring to (b), let $\theta$ be the angle between the momentum of the neutrino and the initial momentum of the pion. Find the value of $\theta$ for which the neutrino's energy is half of its maximum possible energy.

Suzanne W.

Numerade Educator

01:53

Problem 18

A mad scientist asserts to have observed the decay of a particle of mass $M$ into two identical particles of mass $m>0$, with $M<2 m$. He dismisses objections about the violation of the conservation of energy by this process with the claim that if $M$ were traveling fast enough, its energy could easily exceed $2 m c^2$ and could hence decay into the two particles of mass $m$. Prove that he is wrong and qualitatively describe the flaw in his rebuttal.

Chai Santi

Numerade Educator

02:31

Problem 19

Show that a photon cannot spontaneously decay into a particle with a nonzero rest mass, accompanied by an arbitrary number of other particles of arbitrary masses, which may be zero.

Ajay Singhal

Numerade Educator

02:27

Problem 20

In frame $\mathrm{S}$, a photon of frequency $f$ is incident on an electron with momentum $\boldsymbol{p}_1$ and energy $E_1$. Determine the minimum frequency of the scattered photon and the directions of the velocities of the incident photon and the scattered photon in such a situation. The rest mass of the electron is constant.

Keshav Singh

Numerade Educator

02:03

Problem 21

Consider the reaction
$$
\gamma+p \rightarrow p+\pi
$$
The rest energies of a proton and pion are $938 \mathrm{MeV}$ and $135 \mathrm{MeV}$ respectively.
(a) If the proton is initially at rest in the laboratory, find the laboratory threshold photon energy for this reaction to occur.
(b) If the photon's energy is $10^{-3} \mathrm{eV}$, find the minimum proton energy that can spark off this reaction.

Suzanne W.

Numerade Educator

03:32

Problem 22

In the lab frame $\mathrm{S}$, particle 1 of rest mass $m_1$ is traveling at velocity $u$ in the positive $\mathrm{x}$-direction and subsequently collides with an initially stationary particle 2 of rest mass $m_2$. If the final velocities of the particles are still aligned with the $\mathrm{x}$-direction and their rest masses remain constant, find the final energy of particle 2. Determine the fraction of the total energy in the lab frame that is possessed by particle 2 in the limit where $u$ tends to $c$. Hint: Consider the center-of-momentum frame. You will discover the ratio in fact tends to unity. Here's another subtler way of proving this. Let $P_1^{\prime}$ and $P_2$ be the final four-momentum of particle 1 and the initial four-momentum of particle 2 in the center-of-momentum frame. Firstly, show that the inner product of $P_1^{\prime}-P_2$ with itself is larger than zero in the center-of-momentum frame (be wary that $P_1^{\prime}-P_2$ is not a four-vector and its inner product is not invariant). Armed with this inequality, show that the final energy of particle 1 must not exceed $\frac{m_1^2+m_2^2}{2 m_2}$. This value is independent of $u$ and hence shows that particle 2 absorbs most of the energy if $u$ is large.

James Kiss

Numerade Educator

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Problem 23

In inertial frame $\mathrm{S}$, a particle of rest mass $m$ is incident at a speed $u$ on a nucleus of rest mass $M$. During the collision, a photon is emitted. The rest masses of the particle and the nucleus remain unchanged. Show that the maximum energy of the photon in frame $\mathrm{S}$ is
$$
h f=\frac{M m\left(\gamma_u-1\right)}{M+\gamma_u m(1-u)} .
$$
Show that this occurs when the direction of the photon is parallel to the initial velocity of the particle $\boldsymbol{u}$ and when the nucleus and the particle "stick together" after the collision in frame S. Hint: from the four-vector equation $P_1+P_2=P_1^{\prime}+P_2^{\prime}+P_p$, where $P_1, P_2, P_1^{\prime}, P_2^{\prime}$ are the four-momenta of the particle and the nucleus before and after the collision and $P_p$ is the fourmomentum of the photon, obtain $P_1+P_2-P_p=P_1^{\prime}+P_2^{\prime}$ and consider the squared norm of both sides.

Lainey Roebuck

Numerade Educator

01:21

Problem 24

Two particles of rest masses $m$ and $M$ are connected by a flexible rope of constant tension $T$. If the two particles are initially at rest and are separated by an initial distance $L$ in frame $\mathrm{S}$, what is the distance $x$ between the point at which the particles meet and the initial position of the particle of rest mass $m$ ? ("Introduction to Mechanics" by David Morin)

Prabhu Ramji

Numerade Educator

02:20

Problem 25

A particle initially possesses an $\mathrm{x}$-component of momentum $p_0$ and energy $E_0$ in the lab frame. It is then acted upon by a constant $-F$ force in the $y$-direction. Determine the trajectory of the particle in the lab frame. Show that the resultant expression reduces to the familiar parabolic path in the non-relativistic limit.

Vivek Singh

Numerade Educator

01:25

Problem 26

In frame $\mathrm{S}$, there is an initially stationary bucket with zero initial rest mass that is pulled along by a string of constant tension $T$. The bucket begins to gather sand of linear mass density $\lambda$. Find the velocity $u$ of the bucket. ("Introduction to Mechanics")

Dading Chen

Numerade Educator

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Problem 27

In the lab frame $\mathrm{S}$, a source emits light of frequency $f$ towards a glass block of refractive index $n$ that is retracting at velocity $v$. By considering the fourwave vector and the rest frame of the block S', determine the frequency and wavelength of light inside the block in the lab frame $\mathrm{S}$.

Darshan Maheshwari

Numerade Educator

01:05

Problem 28

In the lab frame $\mathrm{S}$, there are uniform electric and magnetic fields $\boldsymbol{E}=$ $(0, E, 0)$ and $\boldsymbol{B}=(0,0, B)$ where $E<0$ and $B>0$. A charged particle of rest mass $m$ and charge $q$ initially possesses velocity $v_0$ in the $\mathrm{x}$-direction when it is at the origin. Determine the maximum $x$-coordinate that the particle attains in its subsequent motion. Hint: Consider another inertial frame.

Prem Bijarniya

Numerade Educator

01:59

Problem 29

In this problem, we shall see how the magnetic field is the manifestation of the electric field in another frame. In frame S, an infinitely long wire carries a current $I$ in the positive x-direction. The current comprises electrons of linear charge density $-\lambda$ traveling at a velocity $u$ in the negative $\mathrm{x}$-direction in frame $\mathrm{S}(I=\lambda u)$. In frame $\mathrm{S}$, there are also stationary positive ions of linear charge density $\lambda$ in the wire such that the wire is neutral. A point charge $q$ travels at a velocity $\boldsymbol{v}$ in the positive $\mathrm{x}$-direction, at a distance $r$ from the wire. Without any knowledge of the existence of a magnetic field and by considering the electric field in the rest frame of the charge S', show that the charge experiences a force of the form
$$
f=q v \times A
$$
in frame $\mathrm{S}$ where $\boldsymbol{A}$ is a certain vector. Now let $\boldsymbol{A}$ be defined as the magnetic field $\boldsymbol{B}$. Verify that the expression for the magnetic field is consistent with that obtained from Ampere's law in frame S. Do not use the transformations for the electromagnetic fields in this problem.

Dominador Tan

Numerade Educator