# 5.9 Relativistic energy  (Page 9/16)

 Page 9 / 16

(a) How long does it take the astronaut in [link] to travel 4.30 ly at $0.99944c$ (as measured by the earthbound observer)? (b) How long does it take according to the astronaut? (c) Verify that these two times are related through time dilation with $\gamma =30.00$ as given.

a. 4.303 y to four digits to show any effect; b. 0.1434 y; c. $1\text{/}\sqrt{\left(1-{v}^{2}\text{/}{c}^{2}\right)}=29.88.$

(a) How fast would an athlete need to be running for a 100- $\text{m}$ race to look 100 yd long? (b) Is the answer consistent with the fact that relativistic effects are difficult to observe in ordinary circumstances? Explain.

(a) Find the value of $\gamma$ for the following situation. An astronaut measures the length of his spaceship to be 100 m, while an earthbound observer measures it to be 25.0 m. (b) What is the speed of the spaceship relative to Earth?

a. 4.00; b. $v=0.867c$

A clock in a spaceship runs one-tenth the rate at which an identical clock on Earth runs. What is the speed of the spaceship?

An astronaut has a heartbeat rate of 66 beats per minute as measured during his physical exam on Earth. The heartbeat rate of the astronaut is measured when he is in a spaceship traveling at 0.5 c with respect to Earth by an observer (A) in the ship and by an observer (B) on Earth. (a) Describe an experimental method by which observer B on Earth will be able to determine the heartbeat rate of the astronaut when the astronaut is in the spaceship. (b) What will be the heartbeat rate(s) of the astronaut reported by observers A and B?

a. A sends a radio pulse at each heartbeat to B, who knows their relative velocity and uses the time dilation formula to calculate the proper time interval between heartbeats from the observed signal. b. $\left(66\phantom{\rule{0.2em}{0ex}}\text{beats/min}\right)\sqrt{1-{v}^{2}\text{/}{c}^{2}}=57.1\phantom{\rule{0.2em}{0ex}}\text{beats/min}$

A spaceship (A) is moving at speed c/ 2 with respect to another spaceship (B). Observers in A and B set their clocks so that the event at ( x, y, z, t ) of turning on a laser in spaceship B has coordinates (0 , 0 , 0 , 0) in A and also (0 , 0 , 0 , 0) in B. An observer at the origin of B turns on the laser at $t=0$ and turns it off at $t=\tau$ in his time. What is the time duration between on and off as seen by an observer in A?

Same two observers as in the preceding exercise, but now we look at two events occurring in spaceship A. A photon arrives at the origin of A at its time $t=0$ and another photon arrives at $\left(x=1.00\phantom{\rule{0.2em}{0ex}}\text{m},0,0\right)$ at $t=0$ in the frame of ship A. (a) Find the coordinates and times of the two events as seen by an observer in frame B. (b) In which frame are the two events simultaneous and in which frame are they are not simultaneous?

a. first photon: $\left(0,0,0\right)$ at $t={t}^{\prime };$ second photon:
$\begin{array}{ccc}t\prime \hfill & =\hfill & \frac{-vx\text{/}{c}^{2}}{\sqrt{1-{v}^{2}\text{/}{c}^{2}}}=\frac{-\left(c\text{/}2\right)\left(1.00\phantom{\rule{0.2em}{0ex}}\text{m}\right)\text{/}{c}^{2}}{\sqrt{0.75}}=-\frac{0.577\phantom{\rule{0.2em}{0ex}}\text{m}}{c}=1.93\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-9}\phantom{\rule{0.2em}{0ex}}\text{s}\hfill \\ \hfill x\prime & =\hfill & \frac{x}{\sqrt{1-{v}^{2}\text{/}{c}^{2}}}=\frac{1.00\phantom{\rule{0.2em}{0ex}}\text{m}}{\sqrt{0.75}}=1.15\phantom{\rule{0.2em}{0ex}}\text{m}\hfill \end{array}$
b. simultaneous in A, not simultaneous in B

Same two observers as in the preceding exercises. A rod of length 1 m is laid out on the x -axis in the frame of B from origin to $\left(x=1.00\phantom{\rule{0.2em}{0ex}}\text{m},0,0\right).$ What is the length of the rod observed by an observer in the frame of spaceship A?

An observer at origin of inertial frame S sees a flashbulb go off at $x=150\phantom{\rule{0.2em}{0ex}}\text{km},y=15.0\phantom{\rule{0.2em}{0ex}}\text{km},$ and $z=1.00\phantom{\rule{0.2em}{0ex}}\text{km}$ at time $t=4.5\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-4}\text{s}.$ At what time and position in the S $\prime$ system did the flash occur, if S $\prime$ is moving along shared x -direction with S at a velocity $v=0.6c?$

$\begin{array}{ccc}\hfill t\prime & =\hfill & \frac{t-vx\text{/}{c}^{2}}{\sqrt{1-{v}^{2}\text{/}{c}^{2}}}=\frac{\left(4.5\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-4}\text{s}\right)-\left(0.6c\right)\left(\frac{150\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}\phantom{\rule{0.2em}{0ex}}\text{m}}{{c}^{2}}\right)}{\sqrt{1-{\left(0.6\right)}^{2}}}\hfill \\ & =\hfill & 1.88\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-4}\phantom{\rule{0.2em}{0ex}}\text{s}\hfill \\ \hfill x\prime & =\hfill & \frac{x-vt}{\sqrt{1-{v}^{2}\text{/}{c}^{2}}}=\frac{150\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}\phantom{\rule{0.2em}{0ex}}\text{m}-\left(0.60\right)\left(3.00\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{8}\phantom{\rule{0.2em}{0ex}}\text{m/s}\right)\left(4.5\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-4}\phantom{\rule{0.2em}{0ex}}\text{s}\right)}{\sqrt{1-{\left(0.6\right)}^{2}}}\hfill \\ & =\hfill & -1.01\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{5}\phantom{\rule{0.2em}{0ex}}\text{m}=-101\phantom{\rule{0.2em}{0ex}}\text{km}\hfill \\ \hfill y& =\hfill & y\prime =15\phantom{\rule{0.2em}{0ex}}\text{km}\hfill \\ \hfill z& =\hfill & z\prime =1\phantom{\rule{0.2em}{0ex}}\text{km}\hfill \end{array}$

#### Questions & Answers

plot a graph of MP against tan ( Angle/2) and determine the slope of the graph and find the error in it.
Ime Reply
expression for photon as wave
BARISUA Reply
Are beta particle and eletron are same?
Amalesh Reply
yes
mari
how can you confirm?
Amalesh
sry
Saiaung
If they are same then why they named differently?
Amalesh
because beta particles give the information that the electron is ejected from the nucleus with very high energy
Absar
what is meant by Z in nuclear physic
Shubhu Reply
atomic n.o
Gyanendra
no of atoms present in nucleus
Sanjana
Note on spherical mirrors
Shamanth Reply
what is Draic equation? with explanation
M.D Reply
what is CHEMISTRY
trpathy Reply
it's a subject
Akhter
it's a branch in science which deals with the properties,uses and composition of matter
Eniabire
what is a Higgs Boson please?
FRANKLINE Reply
god particles is know as higgs boson, when two proton are reacted than a particles came out which is used to make a bond between than materials
M.D
bro little abit getting confuse if i am wrong than please clarify me
M.D
the law of refraction of direct current lines at the boundary between two conducting media of
BATTULA Reply
what is the black body of an ideal radiator
Areej Reply
uncertainty principles is applicable to
Areej
fermions
FRANKLINE
what is the cause of the expanding universe?
FRANKLINE
microscopic particles or gases
Areej
Astronomers theorize that the faster expansion rate is due to a mysterious, dark force that is pulling galaxies apart. One explanation for dark energy is that it is a property of space.
Areej
Thanks for your contribution Areej.
FRANKLINE
no problem
Areej
what is photoelectric equation
HIMANSHU Reply
How does fringe intensity depend upon slit width in single slit diffraction?
Abhishek Reply
intensity seems to be directly proportional radius of slit
Mathieu
what are the applications of Bernoulli's equation
Shaukat
VOLTE
Md Reply
what is Draic equation
M.D
about nuclear angular momentum
rahul Reply
what is spin
MUKESH Reply

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 By By Mistry Bhavesh