HOW FAST IS THAT MOVING?

Hiya, Sean Lally Physics Guy here. Today on “How do we know that?” we are looking at cosmic speeds – aka, how fast is that thing moving?

So, how DO you measure the speed of something? Well, let’s first agree on what speed is, shall we? Speed is the rate at which something covers a particular distance. For our purposes, we will not be splitting hairs of speed versus velocity. Velocity is very similar to speed, but it specifies a direction as well.

Back to speed (represented by a v):

Speed = distance / time

v = d / t

The units are usually something convenient like miles per hour, meters per second, kilometers per hour, furlongs per fortnight….

Let’s say you want to know how fast a car is moving? How might you measure this? Talk to your partner and come up with at least two possible ways to determine the car’s speed.

You probably came up with at least one method that involved a car moving a predetermined distance and you using a stopwatch. Imagine that the car is too far away to get a good look at it. Cars seem big when you’re close to them, but the further you are from the, the smaller they appear until finally they disappear from your sight. Really, if you’re a kilometer from the car, you won’t see it at all.

You might think that using a telescope or binoculars would help out here, and certainly they would up to a point. Of course, it would be really hard to measure the distance that the car is moving. After all, the further away you are, the shorter the car’s traveled distance seems to be (and, if you’re trying the technique learned earlier, the harder to measure the angle). You may have noticed this if you’ve ever watched an airplane pass overhear, or have looked at the ground below as you flew in an airplane. It really doesn’t seem as though you’re moving great distances in short times – in other words, it is hard to get a good estimate of how fast you are moving.

So, is there another way to measure speed?

Glad you asked. Why, yes there is, and it is a fantastic way. Let us consider one of “Physics Greatest Hits”, the Doppler Effect. Let’s experiment a little.

At this point, you will need one or both of the following:

Foam ball (or wiffle ball, cut open and packed with foam) with battery-powered buzzer inside

Tuning fork on a string – it’s really important that the tuning fork be securely fastened to the string; the fork should have a hole in the handle. The amount of string doesn’t matter too much – a meter or so. You will also need a tuning fork mallet to strike it.

You’ll also need a friend nearby. (But then, don’t you always?)

Let’s try out the foam ball. Turn on the buzzer. How would you describe the tone? Now, you are going to throw the ball to your friend. Listen carefully to the sound, and ask your friend to do the same. Throw and listen!

What do you notice? Have your friend throw it back to you and pay careful attention yet again.

You can also try this with the tuning fork. It should have a piece of string tightly attached to the handle. Seriously, a flying tuning fork can be a pretty dangerous thing! Strike the tuning fork and listen to the tone. Strike it again and swing it in a circle above your head. What do you notice? What does your friend notice? Does this matter whether or not the tuning fork is moving toward or away from your friend? Try it again until you notice any trend.

So, something is happening. What is it? Discuss with your friend. Ask other classmates to make sure that everyone agrees.

Change things up a bit. Does the speed of the ball or tuning fork make a difference? Have a third person stand somewhere to pay close attention. Write down all of your observations.

Let’s look at the details of what is really happening.

Think about this. Imagine that your friend is sitting on a chair singing a lovely note: “Laaaaaaaaaaaaaaaaaaaaaaaa…….” If you could actually SEE the note (as a wave), it might resemble this:

(See image 1 above.)

What’s going on here? Well, your friend (the happy face above) is sending out waves with a constant frequency (related to the pitch they are singing). Each wave is expanding like a sphere around your friend, just like good waves do. Let’s imagine that they are sending out 440 waves per second – that’s what musicians call the note ‘A’. Note that wherever you stand, at the points A, B, C or D, you will receive the same number of waves per second (440). In other words, you both agree on the frequency of the note. Only 3 waves are depicted above, the principle is the same.

Now let’s make it trickier – hooray! Imagine that your friend is moving to the right. Maybe they’re on a skateboard zipping by you. Things may get a little weird. Let’s see…

(See image 2 above.)

Things are a bit different now. Your friend is STILL sending out 440 waves per second, but she is moving to the right – that means that each new wave will start at a new location, to the right of where the previous one was released. Therefore, what you receive will depend on where you stand. Note that if you stand at point C, you will receive MORE than 440 waves per second (a higher pitch/frequency). At point B, less than 440 waves per second (a lower pitch/frequency). What your friend sings doesn’t change, but your reception of it does. And it’s not an illusion – you actually DO receive more or fewer waves per second, depending on where you stand. This phenomenon is called the Doppler Effect. You have noticed it for sound, but it also applies to light (and all electromagnetic radiation).

First, let’s talk about sound. You probably noticed that the pitch of sound changed as the ball (or tuning fork) moved. When it moved toward you, the pitch went up. When it moved away from you, the pitch went down.

The tuning fork may have been a bit stranger, as it was moving faster and in a circular path. You should have noticed that the pitch warbled up and down, depending on where it was in its path. Here’s a question - if you swing the fork in a circle, directly above your head, should you notice any effect? Why or why not? Try it and see?

And indeed, this is exactly the case with light or any electromagnetic radiation. If the emitter is moving away from you (relatively speaking), you will receive a lower frequency than you would otherwise. In the physics and astronomy biz, we call this a RED SHIFT.

Similarly, if the emitter is moving toward from you (relatively speaking), you will receive a higher frequency than you would otherwise. We call this a BLUE SHIFT.

By the way, “relatively speaking” refers to the motion between the emitter and the receiver. If you (the receiver) are moving toward the emitter, it is essentially the same as the emitter moving toward you. Of course, if you both move toward (or away from) each other, the effect is even greater.

This does NOT mean that the objects will appear red or blue – rather, a frequency is shifted lower (red shift) if the object is moving away from you. The term ‘red shift’ is used since red is at the low end of the visible spectrum. ‘Blue shift’ is used since blue is near the upper end of the visible spectrum. Incidentally, the term ‘violet shift’ is sometimes used in astronomy.

So perhaps you see where I’m going with this…. Could there be a mathematical way to represent speed as a function of wavelength change?

Oh my, yes.

f’ = f (v +- vd) / (v -+ vs)

Crazy formula, eh? Let’s figure out what these terms mean.

f’ Frequency measured by receiver
f Frequency actually emitted by emitter
v speed of sound (approximately 340 m/s in air)
vd speed of detector
vs speed of source/emitter

How about this +- or -+ business? This is a matter of personal preference for sign convention. I remember the word ‘TA TA,' standing for “toward-away toward-away”. If the detector is moving toward the source, use a plus sign. If the detector is moving away from the source, use a minus sign. In the denominator, it is reversed. If the source is moving toward the detector, use a minus sign. If the source is moving away from the detector, use a plus sign. Got it?

How about the expression for light?

f = fo sqrt [(1 – β)/(1 + β)]

where β = v/c, and sqrt means “square root”. Sign convention is a bit tricky here, too. The above formula is correct for the detector moving away from the source of light. If the detector is moving toward the source, you must reverse both signs in front of β.

For astronomy, an easier formula is often used:

v = ( | Δλ | / λo ) c

| Δλ | the absolute value (positive) of the change in wavelength
λo the emitted wavelength
c the speed of light

By the way, if you only know the frequency and not wavelength, it is easy to determine the wavelength using the simple relationship:

c = f λ

That is, speed of light = frequency x wavelength.

So to get back to our initial question – how can we know how fast something is moving, especially in the case when conventional means are just not practical? We compare a particular wavelength from the object’s spectrum to what we expect from a laboratory standard. Pop the numbers into the formula above and there you go – instant speed!

Further Reading

A good mathematical analysis of the Doppler Effect can be found in most college-level physics texts. Here is one recommendation:

Fundaments of Physics, 8th edition – Halliday, Resnick & Walker. See chapter 17.

ALL TEXT AND IMAGES COPYRIGHT SEAN LALLY 2009 (except where noted)