Relativistic Addition of Velocities *

Transcription

1 OpenStax-CNX module: m Relativisti Addition of Veloities * OpenStax This work is produed by OpenStax-CNX and liensed under the Creative Commons Attribution Liense 3.0 Abstrat Calulate relativisti veloity addition. Explain when relativisti veloity addition should be used instead of lassial addition of veloities. Calulate relativisti Doppler shift. * Version 1.6: Sep 12, :54 pm

2 OpenStax-CNX module: m Figure 1: The total veloity of a kayak, like this one on the Deereld River in Massahusetts, is its veloity relative to the water as well as the water's veloity relative to the riverbank. (redit: abkfenris, Flikr) If you've ever seen a kayak move down a fast-moving river, you know that remaining in the same plae would be hard. The river urrent pulls the kayak along. Pushing the oars bak against the water an move the kayak forward in the water, but that only aounts for part of the veloity. The kayak's motion is an example of lassial addition of veloities. In lassial physis, veloities add as vetors. The kayak's veloity is the vetor sum of its veloity relative to the water and the water's veloity relative to the riverbank. 1 Classial Veloity Addition For simpliity, we restrit our onsideration of veloity addition to one-dimensional motion. Classially, veloities add like regular numbers in one-dimensional motion. (See Figure 2.) Suppose, for example, a girl is riding in a sled at a speed 1.0 m/s relative to an observer. She throws a snowball rst forward, then bakward at a speed of 1.5 m/s relative to the sled. We denote diretion with plus and minus signs in one dimension; in this example, forward is positive. Let v be the veloity of the sled relative to the Earth, u the veloity of the snowball relative to the Earth-bound observer, and u the veloity of the snowball relative to the sled.

3 OpenStax-CNX module: m Figure 2: Classially, veloities add like ordinary numbers in one-dimensional motion. Here the girl throws a snowball forward and then bakward from a sled. The veloity of the sled relative to the Earth is v=1.0 m/s. The veloity of the snowball relative to the truk is u, while its veloity relative to the Earth is u. Classially, u=v+u. : u=v+u (2) Thus, when the girl throws the snowball forward, u = 1.0m/s + 1.5m/s = 2.5m/s. It makes good intuitive sense that the snowball will head towards the Earth-bound observer faster, beause it is thrown forward from a moving vehile. When the girl throws the snowball bakward, u = 1.0m/s + ( 1.5m/s) = 0.5m/s. The minus sign means the snowball moves away from the Earth-bound observer.

4 OpenStax-CNX module: m Relativisti Veloity Addition The seond postulate of relativity (veried by extensive experimental observation) says that lassial veloity addition does not apply to light. Imagine a ar traveling at night along a straight road, as in Figure 3. If lassial veloity addition applied to light, then the light from the ar's headlights would approah the observer on the sidewalk at a speed u=v+. But we know that light will move away from the ar at speed relative to the driver of the ar, and light will move towards the observer on the sidewalk at speed, too. Figure 3: Aording to experiment and the seond postulate of relativity, light from the ar's headlights moves away from the ar at speed and towards the observer on the sidewalk at speed. Classial veloity addition is not valid. : Either light is an exeption, or the lassial veloity addition formula only works at low veloities. The latter is the ase. The orret formula for one-dimensional relativisti veloity addition is u = v + u 1 + vu, (3) where v is the relative veloity between two observers, u is the veloity of an objet relative to one observer, and u is the veloity relative to the other observer. (For ease of visualization, we often hoose to measure u in our referene frame, while someone moving at v relative to us measures u.) Note that the term vu beomes very small at low veloities, and u = v+u gives a result very lose to lassial veloity addition. As before, we see that lassial veloity addition is an exellent approximation to the orret relativisti formula for small veloities. No wonder that it seems orret in our experiene. Example 1: Showing that the Speed of Light towards an Observer is Constant (in a Vauum): The Speed of Light is the Speed of Light Suppose a spaeship heading diretly towards the Earth at half the speed of light sends a signal to

5 OpenStax-CNX module: m us on a laser-produed beam of light. Given that the light leaves the ship at speed as observed from the ship, alulate the speed at whih it approahes the Earth. Figure 4 Strategy Beause the light and the spaeship are moving at relativisti speeds, we annot use simple veloity addition. Instead, we an determine the speed at whih the light approahes the Earth using relativisti veloity addition. Solution 1. Identify the knowns. v=0.500; u = 2. Identify the unknown. u 3. Choose the appropriate equation. u = v+u 4. Plug the knowns into the equation. u = v+u = (0.500)() = ( ) = = = (4) Disussion

6 OpenStax-CNX module: m Relativisti veloity addition gives the orret result. Light leaves the ship at speed and approahes the Earth at speed. The speed of light is independent of the relative motion of soure and observer, whether the observer is on the ship or Earth-bound. Veloities annot add to greater than the speed of light, provided that v is less than and u does not exeed. The following example illustrates that relativisti veloity addition is not as symmetri as lassial veloity addition. Example 2: Comparing the Speed of Light towards and away from an Observer: Relativisti Pakage Delivery Suppose the spaeship in the previous example is approahing the Earth at half the speed of light and shoots a anister at a speed of (a) At what veloity will an Earth-bound observer see the anister if it is shot diretly towards the Earth? (b) If it is shot diretly away from the Earth? (See Figure 5.) Figure 5 Strategy Beause the anister and the spaeship are moving at relativisti speeds, we must determine the speed of the anister by an Earth-bound observer using relativisti veloity addition instead of simple veloity addition. Solution for (a) 1. Identify the knowns. v=0.500; u = Identify the unknown. u 3. Choose the appropriate equation. u= v+u

7 OpenStax-CNX module: m Plug the knowns into the equation. u = v+u = (0.500)(0.750) = (5) = Solution for (b) 1. Identify the knowns. v = 0.500; u = Identify the unknown. u 3. Choose the appropriate equation. u = v+u 4. Plug the knowns into the equation. u = v+u = ( 0.750) 1+ (0.500)( 0.750) = (5) = Disussion The minus sign indiates veloity away from the Earth (in the opposite diretion from v), whih means the anister is heading towards the Earth in part (a) and away in part (b), as expeted. But relativisti veloities do not add as simply as they do lassially. In part (a), the anister does approah the Earth faster, but not at the simple sum of The total veloity is less than you would get lassially. And in part (b), the anister moves away from the Earth at a veloity of 0.400, whih is faster than the you would expet lassially. The veloities are not even symmetri. In part (a) the anister moves faster than the ship relative to the Earth, whereas in part (b) it moves slower than the ship. 3 Doppler Shift Although the speed of light does not hange with relative veloity, the frequenies and wavelengths of light do. First disussed for sound waves, a Doppler shift ours in any wave when there is relative motion between soure and observer.

8 OpenStax-CNX module: m : The observed wavelength of eletromagneti radiation is longer (alled a red shift) than that emitted by the soure when the soure moves away from the observer and shorter (alled a blue shift) when the soure moves towards the observer. 1 + u =λ obs =λ s 1 u. (5) In the Doppler equation, λ obs is the observed wavelength, λ s is the soure wavelength, and u is the relative veloity of the soure to the observer. The veloity u is positive for motion away from an observer and negative for motion toward an observer. In terms of soure frequeny and observed frequeny, this equation an be written 1 u f obs =f s 1 + u. (5) Notie that the and + signs are dierent than in the wavelength equation. : If you are interested in a areer that requires a knowledge of speial relativity, there's probably no better onnetion than astronomy. Astronomers must take into aount relativisti eets when they alulate distanes, times, and speeds of blak holes, galaxies, quasars, and all other astronomial objets. To have a areer in astronomy, you need at least an undergraduate degree in either physis or astronomy, but a Master's or dotoral degree is often required. You also need a good bakground in high-level mathematis. Example 3: Calulating a Doppler Shift: Radio Waves from a Reeding Galaxy Suppose a galaxy is moving away from the Earth at a speed It emits radio waves with a wavelength of m. What wavelength would we detet on the Earth? Strategy Beause the galaxy is moving at a relativisti speed, we must determine the Doppler shift of the radio waves using the relativisti Doppler shift instead of the lassial Doppler shift. Solution 1. Identify the knowns. u=0.825 ; λ s = m 2. Identify the unknown. λ obs 1+ u 3. Choose the appropriate equation. λ obs =λ s 1 u 4. Plug the knowns into the equation. λ obs = λ s 1+ u 1 u = (0.525m) = 1.70 m (5) Disussion Beause the galaxy is moving away from the Earth, we expet the wavelengths of radiation it emits to be redshifted. The wavelength we alulated is 1.70 m, whih is redshifted from the original wavelength of m. The relativisti Doppler shift is easy to observe. This equation has everyday appliations ranging from Doppler-shifted radar veloity measurements of transportation to Doppler-radar storm monitoring. In astronomial observations, the relativisti Doppler shift provides veloity information suh as the motion and distane of stars.

9 OpenStax-CNX module: m : Chek Your Understanding (Solution on p. 12.) Suppose a spae probe moves away from the Earth at a speed It sends a radio wave message bak to the Earth at a frequeny of 1.50 GHz. At what frequeny is the message reeived on the Earth? 4 Setion Summary With lassial veloity addition, veloities add like regular numbers in one-dimensional motion: u=v+u, where v is the veloity between two observers, u is the veloity of an objet relative to one observer, and u is the veloity relative to the other observer. Veloities annot add to be greater than the speed of light. Relativisti veloity addition desribes the veloities of an objet moving at a relativisti speed: u= v+u 1 + vu (5) An observer of eletromagneti radiation sees relativisti Doppler eets if the soure of the radiation is moving relative to the observer. The wavelength of the radiation is longer (alled a red shift) than that emitted by the soure when the soure moves away from the observer and shorter (alled a blue shift) when the soure moves toward the observer. The shifted wavelength is desribed by the equation λ obs =λ s 1 + u 1 u (5) λ obs is the observed wavelength, λ s is the soure wavelength, and u is the relative veloity of the soure to the observer. 5 Coneptual Questions Exerise 2 Explain the meaning of the terms red shift and blue shift as they relate to the relativisti Doppler eet. Exerise 3 What happens to the relativisti Doppler eet when relative veloity is zero? Is this the expeted result? Exerise 4 Is the relativisti Doppler eet onsistent with the lassial Doppler eet in the respet that λ obs is larger for motion away? Exerise 5 All galaxies farther away than about ly exhibit a red shift in their emitted light that is proportional to distane, with those farther and farther away having progressively greater red shifts. What does this imply, assuming that the only soure of red shift is relative motion? (Hint: At these large distanes, it is spae itself that is expanding, but the eet on light is the same.)

10 OpenStax-CNX module: m Problems & Exerises Exerise 6 (Solution on p. 12.) Suppose a spaeship heading straight towards the Earth at an shoot a anister at relative to the ship. (a) What is the veloity of the anister relative to the Earth, if it is shot diretly at the Earth? (b) If it is shot diretly away from the Earth? Exerise 7 Repeat the previous problem with the ship heading diretly away from the Earth. Exerise 8 (Solution on p. 12.) If a spaeship is approahing the Earth at and a message apsule is sent toward it at relative to the Earth, what is the speed of the apsule relative to the ship? Exerise 9 (a) Suppose the speed of light were only 3000 m/s. A jet ghter moving toward a target on the ground at 800 m/s shoots bullets, eah having a muzzle veloity of 1000 m/s. What are the bullets' veloity relative to the target? (b) If the speed of light was this small, would you observe relativisti eets in everyday life? Disuss. Exerise 10 (Solution on p. 12.) If a galaxy moving away from the Earth has a speed of 1000km/s and emits 656 nm light harateristi of hydrogen (the most ommon element in the universe). (a) What wavelength would we observe on the Earth? (b) What type of eletromagneti radiation is this? () Why is the speed of the Earth in its orbit negligible here? Exerise 11 A spae probe speeding towards the nearest star moves at and sends radio information at a broadast frequeny of 1.00 GHz. What frequeny is reeived on the Earth? Exerise 12 (Solution on p. 12.) If two spaeships are heading diretly towards eah other at 0.800, at what speed must a anister be shot from the rst ship to approah the other at as seen by the seond ship? Exerise 13 Two planets are on a ollision ourse, heading diretly towards eah other at A spaeship sent from one planet approahes the seond at as seen by the seond planet. What is the veloity of the ship relative to the rst planet? Exerise 14 (Solution on p. 12.) When a missile is shot from one spaeship towards another, it leaves the rst at and approahes the other at What is the relative veloity of the two ships? Exerise 15 What is the relative veloity of two spaeships if one res a missile at the other at and the other observes it to approah at 0.950? Exerise 16 (Solution on p. 12.) Near the enter of our galaxy, hydrogen gas is moving diretly away from us in its orbit about a blak hole. We reeive 1900 nm eletromagneti radiation and know that it was 1875 nm when emitted by the hydrogen gas. What is the speed of the gas? Exerise 17 A highway patrol oer uses a devie that measures the speed of vehiles by bouning radar o them and measuring the Doppler shift. The outgoing radar has a frequeny of 100 GHz and the returning eho has a frequeny 15.0 khz higher. What is the veloity of the vehile? Note that there are two Doppler shifts in ehoes. Be ertain not to round o until the end of the problem, beause the eet is small.

11 OpenStax-CNX module: m Exerise 18 (Solution on p. 12.) Prove that for any relative veloity v between two observers, a beam of light sent from one to the other will approah at speed (provided that v is less than, of ourse). Exerise 19 Show that for any relative veloity v between two observers, a beam of light projeted by one diretly away from the other will move away at the speed of light (provided that v is less than, of ourse). Exerise 20 (Solution on p. 12.) (a) All but the losest galaxies are reeding from our own Milky Way Galaxy. If a galaxy ly ly away is reeding from us at , at what veloity relative to us must we send an exploratory probe to approah the other galaxy at 0.990, as measured from that galaxy? (b) How long will it take the probe to reah the other galaxy as measured from the Earth? You may assume that the veloity of the other galaxy remains onstant. () How long will it then take for a radio signal to be beamed bak? (All of this is possible in priniple, but not pratial.)

12 OpenStax-CNX module: m Solutions to Exerises in this Module to Exerise (p. 8): Answer f obs =f s 1 u 1 + u Solution to Exerise (p. 10) (a) (b) Solution to Exerise (p. 10) Solution to Exerise (p. 10) a) 658 nm b) red ) v/ = (negligible) Solution to Exerise (p. 10) Solution to Exerise (p. 10) Solution to Exerise (p. 10) Solution to Exerise (p. 11) u =, so u = v+u 1+(vu / ) = (v+) +v v+ 1+(v/ ) = v+ 1+(v/) = = (1.50 GHz) = Solution to Exerise (p. 11) a) b) y ) y (all to suient digits to show eets) Glossary = 1.04 GHz (5) Denition 5: lassial veloity addition the method of adding veloities when v ; veloities add like regular numbers in one-dimensional motion: u = v + u, where v is the veloity between two observers, u is the veloity of an objet relative to one observer, and u is the veloity relative to the other observer Denition 5: relativisti veloity addition the method of adding veloities of an objet moving at a relativisti speed: u= v+u, where v is the relative veloity between two observers, u is the veloity of an objet relative to one observer, and u is the veloity relative to the other observer Denition 5: relativisti Doppler eets a hange in wavelength of radiation that is moving relative to the observer; the wavelength of the radiation is longer (alled a red shift) than that emitted by the soure when the soure moves away

13 OpenStax-CNX module: m from the observer and shorter (alled a blue shift) when the soure moves toward the observer; the shifted wavelength is desribed by the equation λ obs =λ s 1 + u 1 u (5) where λ obs is the observed wavelength, λ s is the soure wavelength, and u is the veloity of the soure to the observer