Chapter 11, Special Relativity: The Lorentz Transformation Video Solutions, Mechanics Berkeley Physics

Problem 1

Lorentz invariant. Verify from Eq. (11.7) that
$$
x^{2}-c^{2} t^{2}=x^{2}-c^{2} t^{2}
$$
Note that if we write $x_{1} \equiv x ; x_{4} \equiv i c t$, then $x^{2}-c^{2} t^{2} \equiv$ $x_{1}^{2}+x_{4}^{2} .$ Here $i=\sqrt{-1}$

Narayan Hari

Numerade Educator

02:08

Problem 2

Lorentz transformation. Given Eq. (11.7), demonstrate Eq. $(11.8)$

Chai Santi

Numerade Educator

01:50

Problem 3

Change of volume. Show that if $L_{0}{ }^{3}$ is the rest volume of a cube, then
$$
L_{0}{ }^{3}\left(1-\beta^{2}\right)^{\frac{1}{2}}
$$
is the volume viewed from a reference frame moving with uniform velocity $\beta$ in a direction parallel to an edge of the cube.

Urvashi Arora

Numerade Educator

07:22

Problem 4

Simultaneity. Show from the Lorentz transformation that two events simultaneous $\left(t_{1}=t_{2}\right)$ at different positions $\left(x_{1} \neq x_{2}\right)$ in reference frame $S$ are not in general simultaneous in reference frame $S^{\prime}$.

Alexander Lorenzo

Numerade Educator

01:37

Problem 5

Change of angle. Calculate in $S^{\prime}$ the length and angle with the $x^{\prime}$ axis of a rod of length $L_{0}$ and angle $\theta$ with the $x$ axis in $S . S^{\prime}$ moves with velocity $V \hat{x}$ with respect to $S$.

Suzanne W.

Numerade Educator

06:31

Problem 6

Addition of velocities. Show that if in the $S^{\prime}$ frame we have $v_{y}^{\prime}=c \sin \theta$ and $v_{x}^{\prime}=c \cos \theta$, then in the $S$ frame
$$
v_{x}{ }^{2}+v_{y}{ }^{2}=c^{2}
$$
The $S^{\prime}$ frame moves with velocity $V \hat{\mathbf{x}}$ with respect to the $S$ frame.

Khoobchandra Agrawal

Numerade Educator

06:34

Problem 7

$\pi^{+}$ mesons
(a) What is the mean life of a burst of $\pi^{+}$ mesons traveling with $\beta=0.73$ ? (The proper mean lifetime $\tau$ is $\left.2.5 \times 10^{-8} \mathrm{~s} .\right) \quad$ Ans. $3.6 \times 10^{-8} \mathrm{~s}$
(b) What distance is traveled at $\beta=0.73$ during one mean life? $\quad$ Ans. $800 \mathrm{~cm}$.
(c) What distance would be traveled without relativistic effects? Ans. $550 \mathrm{~cm}$.
(d) Answer parts $(a)$ to $(c)$ again, but for $\beta=0.99$.

Moaz Ali

Numerade Educator

06:34

Problem 8

$\mu$ mesons. The proper mean life of the $\mu$ meson is approximately $2 \times 10^{-6} \mathrm{~s}$. Suppose that a large burst of $\mu$ mesons produced at some height in the atmosphere travels downward at $v=0.99 c .$ The number of collisions in the atmosphere on the way down is small.
(a) If 1 percent of those in the original burst survive to reach the earth's surface, estimate the original height. [In the $\mu$ meson frame of reference the number of particles which survive to a time $t$ is given by $\left.N(t)=N(0) e^{-t / \tau} .\right]$ Ans. $2 \times 10^{6} \mathrm{~cm}$.
(b) Calculate this distance of travel as measured by the $\mu$ meson.

Moaz Ali

Numerade Educator

03:42

Problem 9

Two events. Consider two inertial frames $S$ and $S^{\prime}$. Let $S^{\prime}$ move with velocity $V \hat{\mathbf{x}}$, with respect to $S .$ At a point $x_{1}^{\prime}$ an event takes place at time $t_{1}^{\prime} .$ At $x_{2}^{\prime}$ another event takes place at time $t_{2}^{\prime} .$ The origins coincide at time $t=t^{\prime}=0 .$ Find the corresponding times and distances in $S$.

Urvashi Arora

Numerade Educator

06:16

Problem 10

$\pi^{+}$ mesons. A burst of $10^{4} \pi^{+}$ mesons travels in a circular path of radius $20 \mathrm{~m}$ at a speed $\beta=0.99 c .$ The proper mean life of the $\pi^{+}$ meson is $2.5 \times 10^{-8} \mathrm{~s}$.
(a) How many survive when the burst returns to the point of origin?
(b) How many mesons would be left in a burst that had remained at rest at the origin for this same period of time?

Linda Winkler

Numerade Educator

03:20

Problem 11

Recessional velocity of galaxy. We stated in Chap. 10 that red-shift data on distant galaxies gave a velocity of recession proportional to distance, in the nonrelativistic region:
$$
V=\alpha r \quad \alpha \approx 1.6 \times 10^{-18} \mathrm{~s}^{-1}
$$
Calculate the recession velocity of a galaxy at a distance of $3 \times 10^{9}$ light yr. Is this velocity relativistic?

Urvashi Arora

Numerade Educator

02:26

Problem 12

Galactic velocities. We observe a galaxy receding in a particular direction at a speed $V=0.3 c$, and another receding in the opposite direction with the same speed. What speed of recession would an observer in one of these galaxies observe for the other galaxy?

Urvashi Arora

Numerade Educator

13:18

Problem 13

Simultaneity. Consider the sources of two events to be located at rest at the points $A$ and $B$, equal distances from the observer $O$ in the frame $S$. Assume that at the particular instant of time (as determined by observer $O$ in $S$ ) at which the two events occur, a second observer $O^{\prime}$ and his associated reference frame $S^{\prime}$, moving with a velocity $V \hat{\mathbf{x}}$ with respect to $S$, coincide with $O$ and his frame $S$ (see Fig. 11.12).
(a) Assume $V / c=\frac{1}{3}$. Sketch the positions of the two frames and the points $A, A^{\prime}, B, B^{\prime}$ when the signal from $B$ arrives at the observer $O^{\prime}$. Has this signal arrived at the observer O? Why?
(b) Sketch the positions of $S$ and $S^{\prime}$ as both signals arrive at $O$.
(c) Sketch the positions of $S$ and $S^{\prime}$ as the signal from $A$ arrives at $O^{\prime}$
(d) Assume that the two events are recorded physically at the points $A^{\prime}, B^{\prime} ;$ for example, on photographic plates. Show under the assumptions of this problem that the distances $A^{\prime} O^{\prime}$ and $B^{\prime} O^{\prime}$ are equal.
(e) Show that the two events are not simultaneous as viewed by $O^{\prime}$. The constancy of the velocity of light under all circumstances is implicitly assumed in the definition of simultaneity. To make this dependence clear consider the following. Let the two events at $A$ and $B$ be the simultaneous radiation of pulses of sound as observed by $O$, an observer at rest with respect to the medium in which the sound is propagated. Let $O^{\prime}$ be an observer moving with a velocity $V$ one-third that of sound.
(f) Use the galilean transformation to show that the velocity of the sound pulses toward $O^{\prime}$ from $A$ and $B$ are not the same as observed by $O^{\prime}$.
(g) Show that even though the two signals arrive at $O^{\prime}$ at different times, the fact that the pulses have traveled with different velocities compensates for this fact and that the two events are inferred to be simultaneous, even by the observer $O^{\prime}$

Robert Zaballa

Numerade Educator

01:31

Problem 14

14. Relativistic doppler shift. Protons are accelerated through a potential of $20 \mathrm{kV}$, after which they drift with constant velocity through a region where neutralization to $\mathrm{H}$ atoms and associated light emission takes place. The $\mathrm{H}_{\beta}$ emission $(\lambda=4861.33 \AA$ for an atom at rest $)$ is observed in a spectrometer. The optical axis of the spectrometer is parallel to the motion of the ions. The spectrum is doppler-shifted because of the motion of the ions in the direction of observed emission. The apparatus also contains a mirror which is placed so as to allow superposition of the spectrum of light emitted in the reverse direction. Recall that $1 \AA \equiv 10^{-8} \mathrm{~cm}$.
(a) What is the velocity of the protons after acceleration? Ans. $2 \times 10^{8} \mathrm{~cm} / \mathrm{s}$
(b) Calculate the first-order doppler shifts, depending on $v / c$, appropriate to the forward and backward directions, and indicate the appearance of the relevant part of the spectrum on a diagram.
(c) Now consider the second-order, or $v^{2} / c^{2}$, effect which arises from relativistic considerations. Show that the second-order shift is $=\frac{1}{2} \lambda\left(v^{2} / c^{2}\right)$, and evaluate this numerically for this problem. Notice that it is the same for both $+\mathbf{v}$ and $-\mathbf{v}$ motions.

Suzanne W.

Numerade Educator