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Current time:0:00Total duration:9:18

Doppler effect formula when source is moving away

Video transcript

in the last video we figured out the the formulas for the observed period and frequency for an observer sitting in the path of the source so the source is moving towards the user observer so this is ik this is the example where the train is moving towards you and you perceive the trains horn as having a higher pitch or a higher frequency and we're able to do that by just kind of doing a thought experiment saying okay my my object starts here after one period a period is just a measure of time but it's the measure of time over which the source emits a cycle so it emits a cycle every period but after one period we said okay where is that first wavefront or that first pulse or that first crest and where is the source because it only exactly one period has passed by and the source will be ready to emit another crest or another cycle so the distance between where the source is and that kind of the front of the crest of that first crest that is going to be the wavelength because this next thing that emits is going to be traveling at the exact same velocity and there's going to be separated by that distance which we saw is this expression we said how long will it take it to travel that distance well it's traveling at a speed of V sub W that'll tell you what the observed period would be for this dude over here we calculated right here and then the observed frequency is just the inverse of that now let's think about the situation where the observer is over here the observer is over here so so this these equations or these formulas that we came up right here this is observer observer or let me say source travelling in direction of the absorb of observer source traveling traveling in observers observers direction now let's think about the opposite case where the source is traveling away from the observer and that's in this case the observer is that guy over there maybe I'll do it in a different color he'll be blue so this is the observer so when we started off our source was right here after exactly one period from the sources point of view the first that first crest emitted has traveled radially outward that our this is the distance the velocity of the wave times the amount of time that's passed use velocity times time is going to give you a distance and where the source of the wave will have traveled to the right exactly this distance it's velocity times the amount of time that's gone by now in the last video we said okay this guy that wave is just passing this guy how long will take for that pulse that's being emitted right then to also reach them and then that tells us the period between two pulses or between two crests and I'll think about that exact same situation here that first crest is just passing this guy and a period of or the T sub s which is a period of the emitted wave has just passed by so this guy is just about to emit another way so that other wave is going to be right here so what is the distance between the crest or the cycle or however you want to think about the pulse that is passing him by right now what is the distance between that and the pulse the front edge of that pulse that is emitting being emitted right at that moment right at that moment well it's going to be this radius which is this value it'll be V sub W times the period that is that distance plus the amount of distance that our source has traveled away from this guy so plus V sub s times the period so that's what this distance is and this is how far apart this wave pulse is going to be from that one or if this crest is going to be from that one so if he's seeing this this first crest right now right at this moment how long will it take him to see the crest that's being emitted right now that's B that's this far away from him well it's that far away from him so let me write this down so it'll take so the amount of time it takes for him to see the next crest or the next the same point in the next cycle that's the period that's the observed period that's going to be equal to this distance the velocity of the wave times the period from the perspective of the source plus the velocity of the source because the source has gotten that much further away from him velocity of the source times the period of the source so that's how far the next crest is and then you divide it by the speed of the wave by the speed of each of the crest which is just the velocity of the wave and we can just factor out the T sub s is here and say this is T sub s times V sub W the velocity of the wave plus the velocity of the source divided by the velocity of the wave so this will be a larger observed period than if this guy was stationary and especially if then if the observer was in the path of the guy that makes sense because every time this guy issues a cycle he is moving a little bit further away so every crest or every the same point in the cycle is going to be further and further apart you're going to have longer wavelengths longer periods and if you want the observed frequency for that guy over there I'll do it in the same color the observed frequency for a guy where the source is traveling away from him is just the inverse of that so one over the period one over the period say market we did there one over the period from the from the point of view of the source is the frequency of the source we've color-coded that is so one over T sub s is equal to the frequency of the source this is the inverse of that so I'm just taking 1 over everything here so 1 over T sub s is the frequency of the source and then we take the inverse of this over here the velocity of the wave divided by the velocity of the wave plus the velocity of the source so we're done at least for the simple cases obviously it becomes a little more interesting when someone isn't exactly in the direction of the source or exactly being moved away from but these are kind of the two extreme cases so this is when this is the situation when it is moving away from you and just to check our math and maybe make it a little bit concrete in relation to the video we did to what we introduced the idea of the Doppler effect let's actually apply those numbers so in that video two videos ago we had a situation where the velocity of our source was 5 meters per second to the right and the velocity of the wave was 10 meters per second outward radially outward and the period of our wave call it the period of our let me do another color the period of our from the point of view of the source was one was one second per cycle and the frequency was just the inverse of that so one cycle per second or one Hertz which is just a cycle per second so using those numbers let's see if we get to the exact same answer we got in that first video where we first learned about the Doppler effect so let's look at the frequency from the point of view of this gentleman right here so the frequency of the source is going to be one one cycle per second one Hertz the velocity of the wave is 10 meters per second let me write this 1 cycle per second velocity wave is 10 meters per second the velocity of the wave is 10 meters per second minus the velocity of the source is 5 meters per second so that what's this going to be equal to the observed frequency for this guy right there the observed frequency for that guy right there's going to be one cycle per second times you have 10 over 10 minus 5 and the and and the meters per second cancel out meters per second the numerator meters per second the denominator so 10 divided by 10 minus 5 or 10 divided by 5 is going to be 2 so it's going to be 2 cycles per second 2 cycles per second and if you want the period the observed period for this guy it's going to be the inverse of that or it's going to be 1/2 seconds per cycle and this is exactly what we got in the previous video or actually those two videos ago now what about the guy who's this guy is running away from well do the exact same thing you have one cycle per second or one Hertz that's the emitted frequency from the point of view of the source times the velocity of the wave divided by the velocity of the wave plus the velocity of the source because moving away from him so it's 10 over 10 plus 5 that's 10 over 15 that is 2/3 so this is equal to 2/3 the units over here I'll cancel out this was cycles per second so 2/3 cycle per second which confirms the numbers we got in that first video so that should make us feel good and it also makes a lot of sense this guy is going to see the wave each each crest more frequently he's going to observe a higher frequency if this a sound a higher pitch this guy since the each crest or the cycles are getting spread out he's going to see them less frequently less frequently and if this is sound he's going to observe a lower pitch