Competitive Physics: Thermodynamics, Electromagnetism and Relativity

Wang Jinhui

Chapter 11 Relativistic Kinematics - all with Video Answers

Educators

Chapter Questions

Problem 1

Adrian shines a laser towards the Moon and forms a red spot on a crater. He claims that if he twists his wrist, the spot on the Moon will travel a great distance in a very short amount of time and thus achieve a superluminal speed - thus violating special relativity. What is wrong with his reasoning?
Now, Betty invents the following thought experiment. Suppose that you build a pair of scissors with very long blades. If you decrease the angle between the handles of the scissors during a certain time interval, the angle between the blades should also decrease by the same amount in the same time interval. Then, points arbitrarily far away from the joint should travel at superluminal speeds as the angular distance covered is fixed! Where does Betty's idea fail?

Jayashree Behera

Numerade Educator

03:56

Problem 2

Muons have a half-life of proper time $t_h$. They are released at a distance $L$ above the surface of the Earth and travel at a constant velocity $v$ towards the Earth. What is the proportion of muons that reach the surface of the Earth? Solve this problem from both the muons' frame and the Earth's frame.

Prabhu Ramji

Numerade Educator

03:59

Problem 3

Consider two frames $\mathrm{S}$ and $\mathrm{S}$ '; $\mathrm{S}$ ' is traveling at a velocity $v$ along the positive $\mathrm{x}$-axis of frame $\mathrm{S}$. A rod, of length $L$ as measured in its rest frame $\mathrm{S}$, subtends an angle $\theta_1$ with the $\mathrm{x}$-axis in frame $\mathrm{S}$. Find the angle subtended by the rod and the x'-axis, $\theta_2$, in frame $\mathrm{S}^{\prime}$.

Zulfiqar Ali

Numerade Educator

05:46

Problem 4

A ladder of proper length, $L$, travels at a relative velocity $v$ towards a barn with proper length $L$. The barn has two doors at its ends that are initially open. From the frame of the barn, frame $S$, the ladder is length-contracted and thus fits into the barn. However, from the frame of the ladder, S', the barn is length-contracted and thus the ladder does not fit. Resolve this apparent paradox. Consider the following: one can show that the ladder fits in the garage by closing the two doors of the barn simultaneously in the current reference frame during the brief period of time that the ladder is completely inside the barn. The doors are then opened to release the ladder when it is about to collide with the doors (so that the doors do not affect the ladder's motion).

Jake Rempel

Numerade Educator

06:55

Problem 5

Consider a spaceship A of proper length $L$ traveling towards an identical spaceship B at a relative velocity $v$. When the right of A reaches the right of B, a cannon is simultaneously fired from the left end of A in S', the frame of spaceship A. In frame S', spaceship B is length-contracted which causes the cannon to miss. However in ship B's frame S, spaceship A is length-contracted and the cannonball seemingly hits. Resolve this apparent paradox. Warning: misleading figure.
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David Morabito

Numerade Educator

01:55

Problem 6

A stationary light source is situated at the origin of frame $\mathrm{S}$. It emits a flash that is received by a receding observer traveling at a velocity $v$ in the positive $\mathrm{x}$-direction. Let the observer's frame be $\mathrm{S}$ '. If $\theta$ and $\theta^{\prime}$ are the angles subtended by the path of the light and the positive $\mathrm{x}$-axis in frame $\mathrm{S}$ and the positive $x$ '-axis in frame S' respectively, show that
$$
\cos \theta^{\prime}=\frac{\cos \theta-\beta}{1-\beta \cos \theta}
$$
where $\beta=\frac{v}{c}$ is defined to be positive in the positive $\mathrm{x}$ or $\mathrm{x}$-direction.
(GRAPH CAN'T COPY)

Prabhu Ramji

Numerade Educator

00:57

Problem 7

Consider two particles traveling at constant velocities $v$ and $u$ in the lab frame. The angle subtended by their velocities is $\theta$. Find the speed of one particle in the frame of the other.

Hast Aggarwal

Numerade Educator

08:17

Problem 8

A train of proper length $L$ is moving longitudinally with velocity $v$ relative to a stationary observer on the ground. A person inside the train, standing at the tail end of the train, throws a ball horizontally with constant velocity $u$ towards the front (assume that there is no gravity) as observed in his own frame, and simultaneously sends a light signal in the same direction. The light hits the front end of the train and is reflected back, meeting the ball at some point. This meeting point of interest is a certain distance from the tail end of the train (as observed in the ground or train's frame).
(a) Find the ratio $R$ of this distance to the proper length of the train $L$ in the train's frame.
(b) In the ground frame, find the ratio $R^{\prime}$ of this distance to the observed length of the train. In your answer, let the length of the train and the velocity of the ball be $L^{\prime}$ and $u^{\prime}$, respectively in the ground frame.
(c) What can you say about your answers in a) and b)? Explain. Hence, derive $u^{\prime}$ in terms of $u$ and $v$.

Khoobchandra Agrawal

Numerade Educator

01:06

Problem 9

Two identical trains are traveling at speeds $\frac{4 c}{5}$ and $\frac{3 c}{5}$ towards the right in frame $\mathrm{S}$. The faster train is initially behind the slower train. Define events $\mathrm{P}$ and $\mathrm{Q}$ to be the front of the faster train crossing the back of the slower train and the back of the faster train crossing the front of the slower train respectively. When event $\mathrm{P}$ occurs in frame $\mathrm{S}$, an observer $\mathrm{R}$ begins walking from the back of the slower train to the front of the slower train. Coincidentally, the time during which he reaches the front of the slower train coincides with event Q. Find the velocity of the faster train in the frame of the slower train. Thus, find the speed of observer $\mathrm{R}$ in the frame S. ("An Introduction to Mechanics")

Nishant Kumar

Numerade Educator

06:31

Problem 10

All velocities in this problem are assumed to be aligned in the $\mathrm{x}$-direction. The rapidity $\phi$ of a particle or frame with respect to a frame $\mathrm{S}$ is defined as
$$
\tanh \phi=\beta=\frac{v}{c}
$$
where $v$ is the velocity of the particle or frame with respect to $\mathrm{S} \cdot \tanh \phi=$ $\frac{e^\phi-e^{-\phi}}{e^\phi+e^{-\phi}}$ is the hyperbolic tangent function. Show that if a particle has rapidity $\phi_1$ with respect to frame 1 and frame 1 has rapidity $\phi_2$ with respect to frame 2 , the particle has rapidity $\phi_1+\phi_2$ with respect to frame 2 . It may be useful to know that $\tanh \left(\phi_1+\phi_2\right)=\frac{\tanh \phi_1+\tanh \phi_2}{1+\tanh \phi_1 \tanh \phi_2}$.
Now, consider a particle which travels at a velocity $v_1$ with respect to frame $S_1$, which travels at velocity $v_2$ with respect to frame $S_2$, which travels at velocity $v_3$ with respect to frame $S_3$, and so on until frame $S_{n-1}$ which travels at velocity $v_n$ with respect to frame $S_n$. All of these velocities are aligned. Show that the velocity of the particle in frame $S_n$ is
$$
u=c \cdot \frac{\prod_{i=1}^N\left(1+\beta_i\right)-\prod_{i=1}^N\left(1-\beta_i\right)}{\prod_{i=1}^N\left(1+\beta_i\right)+\prod_{i=1}^N\left(1-\beta_i\right)}
$$
where $\beta_i=\frac{v_i}{c}$.

Khoobchandra Agrawal

Numerade Educator

03:32

Problem 11

In the lab frame $\mathrm{S}$, a particle is traveling at a velocity $v$ towards an identical, stationary particle. From classical mechanics, we know the resultant velocities of the two particles must be perpendicular after the imminent collision as they have equal masses. Show that it is impossible for the two particles to have non-zero velocities that are perpendicular in special relativity by considering another inertial frame where the situation is symmetric, and assuming that the dynamical laws are reversible. We do not know anything else about the dynamical laws in special relativity now. In fact, you can show that the angle of separation must be smaller than $90^{\circ}$. Finally, prove the classical result, that a head-on collision causes the particles to exchange velocities in frame $\mathrm{S}$, holds in the context of special relativity.

James Kiss

Numerade Educator

01:25

Problem 12

A source that emits photons at frequency $f^{\prime}$ in its own frame, $\mathrm{S}^{\prime}$, is moving across the field of vision of a stationary observer at the origin in the frame of the observer, S. What is the observed frequency of emissions by the source in frame $\mathrm{S}$ ? Now, what is the frequency of the photons emitted at angle $\theta$ in the left diagram below, as seen by the observer when the photons eventually reach his eyes? At the instant where the source is at the closest distance of approach to the observer, what is the frequency of the photons that enter the eyes of the observer? When the observer sees the source at the closest distance of approach, what is the frequency of the photons that enter the observer's eyes? You may find the pictures below to be useful.
(GRAPH CAN'T COPY)

Zachary Warner

Numerade Educator

14:45

Problem 13

Consider the twins' paradox set-up again. Now the two twins send out a radio pulse once per second in their own frames. As before, twin A travels to the distant star, that is a distance $L$ from the Earth in twin B's frame, and back at speed $v$ while twin B remains on the Earth. During the entire process,
(a) How many pulses did twin A broadcast in total?
(b) How many pulses from $\mathrm{B}$ did $\mathrm{A}$ receive in total? Hence, who does twin A conclude to be younger?
(c) How many pulses did twin B broadcast in total?
(d) How many pulses from A did B receive in total? Hence, who does twin B conclude to be younger?

Eduard Sanchez

Numerade Educator

06:00

Problem 14

In lab frame $S$, a stationary source emits light of frequency $f$ in vacuum, in the positive $\mathrm{x}$-direction. The photons then pass through a glass block of refractive index $n$ and proper length $l$ that is traveling at a velocity $v$ in the positive $\mathrm{x}$-direction. Determine the time taken by the light to cross the block, the frequency and wavelength of light inside the block in the frame of the block, $S^{\prime}$. Ditto for the lab frame.

Ceren Uzun

Texas Tech University

06:49

Problem 15

In the lab frame $S$, three lamps at coordinates $x_1, x_2$ and $x_3$ are observed to be illuminated at times $t_1, t_2$ and $t_3$. At $t=0$ in $S$, a car is observed to travel from the origin at a constant velocity $v>0$ in the positive $\mathrm{x}$-direction. Under what conditions will the person $P$ in the car observe all three lamps to be lit up simultaneously? Next, assume that $P$ observes the events to occur at $t^{\prime}=0$ in his own frame $S^{\prime}$. Let the time intervals between the illumination of the lamps and the receipt of the corresponding photons be $\Delta t_1, \Delta t_2$ and $\Delta t_3$ in frame $S^{\prime}$. If person $P$ observes the ratio of these intervals to be $\Delta t_1: \Delta t_2: \Delta t_3=1: 2: 3$, and given $x_1$, determine the $\mathrm{x}$-coordinates of the other lamps in frame $\mathrm{S}$ and the times at which the lights were lit up in frame S. Solve this problem via a Minkowski diagram.

Zulfiqar Ali

Numerade Educator

04:16

Problem 16

In the lab frame, car 1 travels at speed $v_1=\tan 15^{\circ} \mathrm{c}$ in the negative $\mathrm{x}-$ direction while car 2 travels at speed $v_2=\frac{c}{\sqrt{3}}$ in the positive $\mathrm{x}$-direction. The cars start from the origin $O$ at time $t \stackrel{\sqrt{3}}{=} 0$. At a certain later time, car 1 emits a light signal in the positive $\mathrm{x}$-direction. If an observer in car 1 measures the time interval between the emission event and the receiving event by car 2 to be $t^{\prime}$, determine the distance that car 2 has traveled from its initial position in the lab frame when it receives the light signal with the aid of a Minkowski diagram.

Ryan Hood

Numerade Educator

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Problem 17

The velocity of a particle as a function of time $t$ in the lab frame is given by
$$
u(t)=c \sqrt{1-\frac{1}{\left(\frac{g t}{c}+1\right)^2}}
$$
and is oriented along the positive $\mathrm{x}$-axis.
(a) Show that the proper time elapsed in the particle's frame is $\tau=\frac{c}{g} \ln \left(\frac{g t}{c}+1\right)$.
(b) Let $x$ denote the instantaneous $\mathrm{x}$-coordinate of the particle in the lab frame. If the particle starts at the origin in the lab frame originally, show that $x(\tau)=\frac{c^2}{g}\left[\sqrt{e^{\frac{2 g \tau}{c}}-1}-\tan ^{-1} \sqrt{e^{\frac{2 g \tau}{c}}-1}\right]$.

Victor Salazar

Numerade Educator

01:27

Problem 18

In the standard configuration, a particle moves in the $\mathrm{x}$-direction. In the lab frame, its $\mathrm{x}$-coordinate is described by
$$
x(\tau)=\frac{c^2}{g}\left(\cosh \left(\frac{g \tau}{c}\right)-1\right),
$$
where $g$ is a constant with units of acceleration and $\tau$ is the proper time of the object. Define $\gamma_u$ as the gamma factor ascribed to the speed of the object $u$ in the lab frame.
(a) Express $u$ in terms of $\gamma_u, g, c$ and $\tau$.
(b) Hence, express $\gamma_u$ in terms of $g, c$ and $\tau$.
(c) Using the result of (b), re-express $u$ solely in terms of $g, c$ and $\tau$.
(d) Express $t$ as a function of $\tau$ and hence, $u(t)$ and $a(t)$. Show that $u(t)$ and $a(t)$ make sense for $t \rightarrow \infty$.

Hunza Gilgit

Numerade Educator