Doppler Effect (Text 1.3)

Transcription

1 Doppler Effet (et 1.3) Consider a light soure as a soure sending out a tik eery 1/ν and these tiks are traeling forward with speed. tik tik tik tik

2 Doppler Effet (et 1.3) Case 1. Obserer oing transersely. tik tik tik tik Note that is proper tie, so period obsered by will be γ. Hene, the frequeny of the light wae obsered by is gien by ν ν γ ν ν Obsered frequeny is always lower. 1 2

3 Doppler Effet (et 1.3) Case 2. Obserer oing away fro the light soure. tik tik tik tik If is the period in the soure frae. his orrespond to a tie of γ in s eye, but this annot be the period in s frae beause bob is oing away fro the soure at the sae tie. So, if is the period obsered by bob, we hae (+) γ. (1+ ) ( + ) γ (1+ ) γ 1 + ν ν - + Obsered frequeny is lower. Sine is onstant, waelength has to be longer (red shift). hat s how we know other galaies are oing away fro us. 2

4 Doppler Effet (et 1.3) Case 3. Obserer oing towards the light soure. tik tik tik tik 2 1 ) (1- ) (1 ) ( ν ν γ γ + + Obsered frequeny is higher. Sine is onstant, waelength has to be shorter (blue shift). If is the period in the soure frae. his orrespond to a tie of γ in s eye, but this annot be the period in s frae beause bob is oing towards the soure at the sae tie. So, if is the period obsered by bob, we hae (-) γ.

5 Doppler Effet (et 1.3) S S S S 1 ν λ λ λ ν λ λ Classial ase. his apply only to lassial waes, like sound and water waes.. tik tik tik tik If the soure is oing towards and is not oing, the waelength obsered by is S If is oing towards the soure the tiks will approah hi with a speed of + O instead of. So the frequeny obsered by is + O ν ν

6 Doppler Effet (et 1.3) Cobining two results, + ν ν - S and O are positie if the soure and obserer are oing towards eah other, negatie if they are oing away fro eah other. Notes: 1. rouble ours if the soure approah the obserer at the speed of sound! O S 2. Different fro relatiity, s and O are two different eloities in the aboe equation. Only relatie eloity ( s - o ) is iportant in the ase of relatiity. What is the referene frae for S and O? Soue is oing towards left, while obserer is stationary.

7 Doppler Effet (et 1.3) 1. All galaies show red shift all galaies are oing away fro us. 2. he reession speed of a galay is proportion to its distane fro us (Hubble Law). Edwin Hubble 3. he proportional onstant is about 71 ± 4 (k/s)/mp. 1p he unierse is epanding, but it has no enter. 5. More reent disoery: the unierse is epanding faster and faster dark energy.

8 Relatiisti Moentu (et 1.7) Y Y s partile and Nora partile hae the sae ass, but only when it is at rest. s partile ass will be different fro (say, let it be ) in Nora s easureent, beause it is oing in Nora s frae. he purpose of this thinking eperient is to deterine.

9 Relatiisti Moentu (et 1.7) Y Y he eperient: and Nora plan ahead and they know what to do. Before the y-aes of the two inertial frae oinide, push his partile in the y diretion with a speed and Nora push her partile in the +y diretion with the sae speed. We hae to assue to be ery slow so it has no effet on the partile ass. hey plan it so well that when the y-aes oinide, the two partiles ollide!

10 Relatiisti Moentu (et 1.7) What see: Y What Nora see: s partile Y Nora s partile Although the ass appears different in different inertial fraes (i.e. ass is NO an inariant), but onseration of oentu should hold in ALL inertial fraes (postulate I)!

11 Relatiisti Moentu (et 1.7) What Nora see: s partile Y does not equal, beause we hae different tie! Before Nora s analysis, we need to note that: 1. is not the sae as beause and Nora are easuring the sae distane (along the y-aes) with a different lok, so y t 2. he ollision is elasti. y γ t Nora 3. We will assue the ass of a partile depends on its speed only. γ 4. he ass of s partile ay be different beause it ay oe with a different speed (so we add a bar to it in our notation) 5. Just like Nora, will see his partile oing along the y ais in his frae. So s partile has the sae horizontal (-) oponent in Nora s frae, before and after the ollision. Iportant

12 Relatiisti Moentu (et 1.7) What Nora see: Y Now Nora do an analysis based on onseration of oentu: Before ollision : p y y p After ollision : p p p p - ' ' ' - ' his eans 's partile is oing with the sae speed before and after ollision, + + ' ' ' '

13 Relatiisti Moentu (et 1.7) What Nora see: Analysis ontinue: Y p - ' - ' - ' y (beause ' ' and 2 2' ' ' ' ' γ py + ' + ' + ' γ )

14 Relatiisti Moentu (et 1.7) We define the rest ass (or proper ass) as the ass of a partile when it is at rest. If is ery sall, then the in the preious equation is atually and we an rewrite the equation as: γ Note that when. With this newly defined, the oentu of a oing partile should be: p γ 2 1 And Newton s seond law beoes: r r dp F dt d dt 2 1 r ( γ )