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There was a formula that showed cos(theta), and I can see that I missed using it. But on a content review perspective, wouldn't a perpendicular movement still show changes in the doppler effect? I drew a diagram to illustrate my thinking. The distance between the observer and object are still decreasing and increasing as the object moves perpendicular to the observer, even if it's at an angle. Why wouldn't the dopper effect work for a perpendicular mover?
Thank you!
The question does not ask about the doppler shift, but about the PAD frequency shift. As for your perpendicular question, I assume you mean "at a right angle to the motion of the detector"
If so, there will be a Doppler effect as anytime 2 objects are moving relative to each-other you will have a Doppler shift.
However if we mean if the source is moving perpendicular to a line connecting the source and detector, then no, there will be no shift as this means the objects are moving parallel to each other.
If it the motion is initially at a right angle to the detector, and continues in a straight line, then yes there is a Doppler effect because the object is moving away from the observer.
But to be clear on a few things:
1) If it continues in a straight line, the angle is no longer 90 degrees but is instead constantly changing.
2) When moving at a right angle to the detector, it's impossible for it to be approaching the observer, it must be moving away, so the Doppler effect will always be a redshift.
3) If, instead of continuing in a straight line, it circles the observer, the angle will stay at a constant 90 degrees and there will be NO Doppler effect.
Hope this helps, good luck!