# 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 $v=0.6c?$

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

plot a graph of MP against tan ( Angle/2) and determine the slope of the graph and find the error in it.
expression for photon as wave
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