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Explain the meaning of the terms “red shift” and “blue shift” as they relate to the relativistic Doppler effect.
What happens to the relativistic Doppler effect when relative velocity is zero? Is this the expected result?
Is the relativistic Doppler effect consistent with the classical Doppler effect in the respect that ${\lambda}_{\text{obs}}$ is larger for motion away?
All galaxies farther away than about $\text{50}\times {\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}\text{ly}$ exhibit a red shift in their emitted light that is proportional to distance, with those farther and farther away having progressively greater red shifts. What does this imply, assuming that the only source of red shift is relative motion? (Hint: At these large distances, it is space itself that is expanding, but the effect on light is the same.)
Suppose a spaceship heading straight towards the Earth at $0\text{.}\text{750}c$ can shoot a canister at $0\text{.}\text{500}c$ relative to the ship. (a) What is the velocity of the canister relative to the Earth, if it is shot directly at the Earth? (b) If it is shot directly away from the Earth?
(a) $0\text{.}\text{909}c$
(b) $0\text{.}\text{400}c$
Repeat the previous problem with the ship heading directly away from the Earth.
If a spaceship is approaching the Earth at $0.100c$ and a message capsule is sent toward it at $0.100c$ relative to the Earth, what is the speed of the capsule relative to the ship?
$0\text{.}\text{198}c$
(a) Suppose the speed of light were only $\text{3000 m/s}$ . A jet fighter moving toward a target on the ground at $\text{800 m/s}$ shoots bullets, each having a muzzle velocity of $\text{1000 m/s}$ . What are the bullets’ velocity relative to the target? (b) If the speed of light was this small, would you observe relativistic effects in everyday life? Discuss.
If a galaxy moving away from the Earth has a speed of $\mathrm{1000\; km/s}$ and emits $\text{656 nm}$ light characteristic of hydrogen (the most common element in the universe). (a) What wavelength would we observe on the Earth? (b) What type of electromagnetic radiation is this? (c) Why is the speed of the Earth in its orbit negligible here?
a) $\text{658 nm}$
b) red
c) $v/\text{c}=9\text{.}\text{92}\times {\text{10}}^{-5}$ (negligible)
A space probe speeding towards the nearest star moves at $0\text{.}\text{250}c$ and sends radio information at a broadcast frequency of 1.00 GHz. What frequency is received on the Earth?
If two spaceships are heading directly towards each other at $0\text{.}\text{800}c$ , at what speed must a canister be shot from the first ship to approach the other at $0\text{.}\text{999}c$ as seen by the second ship?
$0\text{.}\text{991}c$
Two planets are on a collision course, heading directly towards each other at $0\text{.}\text{250}c$ . A spaceship sent from one planet approaches the second at $0\text{.}\text{750}c$ as seen by the second planet. What is the velocity of the ship relative to the first planet?
When a missile is shot from one spaceship towards another, it leaves the first at $0\text{.}\text{950}c$ and approaches the other at $0\text{.}\text{750}c$ . What is the relative velocity of the two ships?
$-0\text{.}\text{696}c$
What is the relative velocity of two spaceships if one fires a missile at the other at $0.750c$ and the other observes it to approach at $0.950c$ ?
Near the center of our galaxy, hydrogen gas is moving directly away from us in its orbit about a black hole. We receive 1900 nm electromagnetic radiation and know that it was 1875 nm when emitted by the hydrogen gas. What is the speed of the gas?
$0\text{.}\text{01324}c$
A highway patrol officer uses a device that measures the speed of vehicles by bouncing radar off them and measuring the Doppler shift. The outgoing radar has a frequency of 100 GHz and the returning echo has a frequency 15.0 kHz higher. What is the velocity of the vehicle? Note that there are two Doppler shifts in echoes. Be certain not to round off until the end of the problem, because the effect is small.
Prove that for any relative velocity $v$ between two observers, a beam of light sent from one to the other will approach at speed $c$ (provided that $v$ is less than $c$ , of course).
$u\prime \phantom{\rule{0.25em}{0ex}}=c$ , so
$\begin{array}{ll}u& =& \frac{\text{v+u}\prime}{1+(\text{vu}\prime /{c}^{2})}=\frac{\text{v+c}}{1+(\text{vc}/{c}^{2})}=\frac{\text{v+c}}{1+(v/c)}\\ & =& \frac{c(\text{v+c})}{\text{c+v}}=c\end{array}$
Show that for any relative velocity $v$ between two observers, a beam of light projected by one directly away from the other will move away at the speed of light (provided that $v$ is less than $c$ , of course).
(a) All but the closest galaxies are receding from our own Milky Way Galaxy. If a galaxy $\text{12}\text{.}0\times {\text{10}}^{9}\phantom{\rule{0.25em}{0ex}}\text{ly}$ ly away is receding from us at 0. $0.900c$ , at what velocity relative to us must we send an exploratory probe to approach the other galaxy at $0.990c$ , as measured from that galaxy? (b) How long will it take the probe to reach the other galaxy as measured from the Earth? You may assume that the velocity of the other galaxy remains constant. (c) How long will it then take for a radio signal to be beamed back? (All of this is possible in principle, but not practical.)
a) $0\text{.}\text{99947}c$
b) $1\text{.}\text{2064}\times {\text{10}}^{\text{11}}\phantom{\rule{0.25em}{0ex}}\text{y}$
c) $1\text{.}\text{2058}\times {\text{10}}^{\text{11}}\phantom{\rule{0.25em}{0ex}}\text{y}$ (all to sufficient digits to show effects)
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