# SR #2: Relative Motion

In the last two posts I’ve explained how Special Relativity is about relative motion between two frames of reference, and that the motion involved is constant, straight line motion that allows us to view either frame as the “moving” one or the “standing still” one.

Today I’m going to dig a little bit deeper into the idea of relative motion and what that involves for actions within a constantly moving frame of reference versus what observers in a different frame perceive. In other words: trains, planes, and automobiles.

(Warning: this gets a little math-y, but you can ignore those bits.)

Let’s say you’re holding up a bowling ball and you drop it.[1] Exactly as you expect, it falls straight down and hits the floor. (Watch out for your toes!)

It falls very fast, but it does take some amount of time to fall. If we had quick reflexes and a good stopwatch, we could measure the amount of time.

In diagram 1 we see that our friend Al has performed this experiment. The cyan-colored ball shows the starting position, the yellow-colored ball shows the final position. To the right is a stop watch that has measured three ticks (let’s call them millijiffies) for the ball to fall.

Now let’s imagine that our friend Em performs the same experiment. A key tenet of physics is that identical experiments performed in different frames of reference always have identical results.[2]

Diagram 2. Three ticks for Em.

That means that Em gets the same three milli-jiffy result as she times the falling bowling ball (we assume she drops it from the same height — if she didn’t, she would get a different time).

What I didn’t tell you is that Em is actually on a moving train traveling past Al (but remember, from her point of view, she is still, and Al is traveling past her). As you can imagine, since Em perceives herself as standing still (just as Al does), she must get the same result as Al does.

If Al watches Em as she passes, he sees that her stopwatch and his agree. They see that both experiments take three ticks. But consider what Al actually sees as Em passes. We’ll assume Em is moving from Al’s left to his right.

If Em is moving (according to Al), so is the falling ball (and the stopwatch and everything else in Em’s frame of reference). However the ball is not only moving left to right, it’s also moving down! From Al’s point of view, it looks like this:

Diagram 3. What Al sees when Em drops the ball.

According to Al, Em’s falling ball moves in a diagonal path![3] He also sees Em’s stopwatch and knows the ball took three ticks to move along the diagonal path.

If Al and Em both drop their bowling balls from a height of three cubits, they both calculate the ball’s fall rate (its speed) as one cubit per milli-jiffy (simply: 3 cubits ÷ 3 milli-jiffies = 1).

Pythagorean theorem

But as you see in diagram 3, Al also sees that Em’s ball moves six cubits to the right as it falls. He can use the Pythagorean theorem to calculate the total distance it moves: a smidgen over 6.7 cubits.[4]

If Al divides this by the three ticks it takes to fall, he sees that Em’s ball is moving over twice as fast (a hair under 2.24 times) as his.[5]

Now consider what the situation looks like from Em’s point of view as she — considering herself as standing still — watches Al pass her (from right to left from her perspective):

Diagram 4. What Em sees when Al drops the ball.

She sees the same picture of Al that Al saw of her (just with the direction reversed). She would make the same calculations of Al’s experiment that Al made of hers. And they’d both be right.

This is the critical point: Al sees Em’s ball as moving faster (because he thinks Em is moving), and Em sees Al’s ball as moving faster (because she thinks Al is moving). And both are correct!

This is the basic form of relativity known to the ancients (who observed that if you drop a rock while passing by the shore on a fast ship, the same sort of situation obtains as with Al and Em).

This idea extends to our infrequent guest, Max, who happens to be performing the bowling ball experiment on a passing plane:

Diagram 5. Special guest Max and his (rather slow) plane.

Al sees Max’s bowling ball move much faster than Em’s did (because Max is moving much faster).

Em, because she’s also moving (we’ll assume both Em and Max are traveling in the same direction) won’t see Max’s speed as quite as fast as Al does — her speed deducts from Max’s. Yet Max is still moving relative to her, so she sees Max’s ball as moving faster (just not as fast as Al does).

Of course, Max sees his bowling ball fall straight down, exactly as Em and Al see their respective balls falling straight down. He also sees their balls moving faster than his the same way they do his.

And that’s enough for today. Next time we’ll pick up with the idea of relative motion — how speeds combine — and how there turns out to be a critically important exception!

If you are wondering what happened to the automobiles… how do you think Em and Max got to the train station and the airport in the first place?

[1] We’re using bowling balls in order to have a heavy enough object that we can ignore air resistance. Ever since Galileo, we’ve known that — except for air resistance — all objects (on Earth) fall at the exact same speed.

There is a demonstration in a large vacuum chamber where a feather and a bowling ball are dropped simultaneously, and the feather falls just as fast as the bowling ball! Check out the link; it’s amazing!

[2] To be clear, we’re talking about inertial frames of reference — that is, the frames of reference moving at constant, straight-line, speed as we’ve discussed.

[3] Actually, it moves in a parabolic path because the bowling ball starts falling at zero inches per milli-jiffy and falls faster and faster (accelerates!) until — at some point — wind resistance stops it from speeding up (terminal velocity).

But the distances and accelerations involved here are small enough that we can ignore them.

[4] 32 + 62 = 9 + 36 = 45. The square root of 45 is 6.7082 plus some (very small) change.

[5] Em travels six cubits in that three milli-jiffies, so she is moving at a rate of two cubits per milli-jiffy (6 ÷ 3 = 2). That speed combines with the ball’s speed (in a Pythagorean way) to give us the 2.24 result.

Specifically: 12 + 22 = 1 + 4 = 5. The square root of 5 is just under 2.24.

The canonical fool on the hill watching the sunset and the rotation of the planet and thinking what he imagines are large thoughts. View all posts by Wyrd Smythe

#### 25 responses to “SR #2: Relative Motion”

• dianasschwenk

One of my fave pastimes used to be walking up a down escalator – should have tried it while dropping a bowling ball! ❤
Diana xo

Nice diagrams! And I’m once again appreciating the simplicity of your explanations.

• Wyrd Smythe

Thank you, and thank you! I’m glad you like the diagrams; there will be a lot more of them from here on out. (I created them all last January while I was taking a break from blogging. They’ve been waiting in the wings for their big appearance ever since. 🙂 )

• user12262

Wyrd Smythe wrote (March 18, 2015):
> A key tenet of physics is that identical experiments […] always have identical results.

That’s not profound at all, but (boils down to) a banal definition:
if two (distinct) experimental trials had (positively, measured) equal setups and (positively, measured) equal results then we may as well call them two equal experimental trials.

The key issue of (experimental) physics is rather in how to determine (reliably, in mutual agrteement) whether given setups had been equal (e.g. whether “Al” and “Em” really had dropped their respective balls equally “straight down”, and equally “freely”), or by how much their individual trial setups differed; and whether the result values obtained had been equal (e.g. whether “three ticks on Al’s stop watch” did indeed have equal duration as “three ticks on Em’s stop watch“), or by how much they differed.

• Wyrd Smythe

I think you’re missing the point. The isotropy of physics is profoundly important; it’s what justifies us reasonably believing that the physics we can test here on Earth is the same physics that applies to the rest of the universe. The consistency of science depends on physics being isotropic with regard to space and to time.

Al and Em, or just Al or Em, can use the same piece of experimental equipment to insure the experiment is performed identically in two places. Alternately, two devices can be built according to the same plans and tested side-by-side to ensure they give identical results while in the same frame of reference.

• user12262

Wyrd Smythe wrote (March 20th, 2015 at 9:07 pm):
> […] what justifies us reasonably believing that the physics we can test here on Earth is the same physics that applies to the rest of the universe.

You’ve hit exactly the point I’ve been trying to get at; and yes, there’s some profound disagreement (between us, FWIW), or at least need for clarification, regarding what “justifies us reasonably believing [about nature]” and what exactly is meant by “the physics“:

We may “design tests” and “invent measurement algorithms” so that they may in principle be applied likewise anywhere/anytime, and to any suitable set of observational data. (At least we may try and consider this as a goal.) This justifies making comparisons between the result values of different trials; this makes the various result values commensurate.

But there is no particular up-front justification for expecting some particular result value from any one trial; or expecting that result values from different trials ought to be equal, or in some particular relation to each other. This could only justly be concluded once the various results had been obtained separately (and without bias).

Therefore the believe in “physics being isotropic with regard to space and to time” is justified only (at best) as a characterization of the measurement method; but it doesn’t apply up-front to result values which might thereby be obtained. Result values are not contingent on belief, but “on experiment”: on the availability of observational data and on the choice of measurement method (for deriving result values from the available data). And indeed we find that our more or less immediate surroundings are far from isotropic; and whether this is qualitatively different “the farther we look”, or not, is also not a matter of belief, but “of experiment”.

> Al and Em, or just Al or Em, can use the same piece of experimental equipment to […]

Using the same physical piece(s) of equiment in two distinct trials does not say that these were and remained equal in these trials. Al speaking of “three milli-jiffies” and Em speaking of “three milli-jiffies” does not itself guarantee that they are referring to equal durations; even if they readily agree that they mean the same rational number by “three milli” (presumably “0.003”). Instead: it can and must be measured, trial by trial, whether what Al meant by “jiffie” and what Em meant by “jiffie” were equal durations, or how they were related to each other. (You wouldn’t expect either that all your readers would mean and understand the exact same duration by “a jiffie”, would you?)

> devices can be built according to the same plans and tested side-by-side to ensure they give identical results while in the same frame of reference.

Sure. But this doesn’t spare making a plans and carrying out measurements of comparing such devices also while they were not at rest to each other. (And that’s the key to SR, AFAIU.)

• Wyrd Smythe

“Result values are not contingent on belief, but ‘on experiment’: on the availability of observational data and on the choice of measurement method (for deriving result values from the available data).”

Well sure. The history of science involves exactly that. At this point Relativity (Galilean, Special, and General) have all gone through that process. Relativity is one of the most thoroughly tested theories in science.

I’m not sure what point you’re trying to make here. Do you feel I’m asserting matters of belief in this discussion? I’m not; I’m reporting the end result of, literally, centuries of scientific experiments going back to Galileo. This series of articles is about those results and the theory we think accounts for them.

“Using the same physical piece(s) of equiment in two distinct trials does not say that these were and remained equal in these trials.”

Such matters are easily tested. If two identical devices, side-by-side, give identical results for N tests (for any value of N that suits you) are you not justified in saying they are, in fact identical devices?

If Em takes one of them and performs further tests on a moving train — continuing to get the same result in test after test — and then returns to Al — who continued to make further tests and continued to get the same result — and they once again compare side-by-side results — which continue to match — have they not established identical results throughout?

We can even attach accurate time pieces to both and compare the total elapsed time when Em returns her device to Al, thus demonstrating that time has elapsed the same for both. If Em and Al can see each other while performing tests from different frames, they confirm the continued identical behavior of their devices.

“Instead: it can and must be measured, trial by trial, whether what Al meant by ‘jiffie’ and what Em meant by ‘jiffie’ were equal durations, or how they were related to each other.”

Well, sure, but that’s all taken as given (because those things have been tested, and we’re assuming that’s all a given for Al and Em).

If your point is that scientific results are based on testing and measuring, well absolutely they are! No one here has said otherwise.

• user12262

Wyrd Smythe wrote (March 21st, 2015 at 11:18 am):
> If two identical devices, side-by-side, give identical results for N tests (for any value of N that suits you) are you not justified in saying they are, in fact identical devices?

Well: If two devices (regardless how you call them initially) give equal results in N trials (for N > 0) then

– we are justified in saying that these two devices, including their (respective, or joint) “conditions” in these N respective trials, had been equal to each other in these N trials; and

– some may even offer “odds” on the outcome of “trial N+1”.

But this certainly doesn’t spare actually conducting and evaluating this “trial N+1” in any case. And if the “trial N+1” is even explicitly supposed to be qualitatively distinct from the previous N trials (e.g. by the devices being required to be sparated from each other, instead of having been “side-by-side“), then perhaps some might become less confident in their expectations regarding the outcome.

> We can even attach accurate time pieces to both [Al and Em …]

We may certainly ask (you) for a method of determining which of the numerous imaginable “time pieces” should be called (separately) “accurate“.
(This may even involve “returning devices”; even though the experiments sketched in the post above don’t seem to involve “Al and Em returning to each other”.)

> […] what point you’re trying to make here.

Seems, eventually, to call into question your concept of “The history of science“; with its particular application to SR and its (well-known) thought experiments.
(Our brief correspondence so far has certainly be pretty efficient for identifying this substantial disagreement …)

• Wyrd Smythe

“some may even offer “odds” on the outcome of ‘trial N+1’.”

No, everyone would offer odds of some kind. Most would offer very high odds as N increases with consistent results.

Just what odds would you offer against experiments in Galilean invariance returning different answers than they have through centuries of unequivocal test results?

“But this certainly doesn’t spare actually conducting and evaluating this ‘trial N+1’ in any case.”

But that is always true. At what point do you accept your test results as indicative of the underlying physics?

Again: with regard to all forms of relativity, N is extremely large and extremely consistent. Relativity is one of the most tested theories in all of science.

“This may even involve ‘returning devices’; even though the experiments sketched in the post above don’t seem to involve ‘Al and Em returning to each other’.”

They don’t need to. Others — many, many others — have performed these tests during the last century, and the results are consistent and unequivocal.

These posts are about the physics of relativity, not the well-established experimental history. I’m not writing about the experiments, but the results.

Do you disagree with the physics of Galilean invariance (which is all I’ve written about so far)?

“…SR and its (well-known) thought experiments.”

Do you understand that the tests of SR go far beyond thought experiments? Again: one of the most tested theories in science.

• user12262

Wyrd Smythe wrote (March 22nd, 2015 at 9:51 pm):
> […] Do you disagree with the physics of Galilean invariance (which is all I’ve written about so far)?

Again:
Do you claim that everyone, outright, means exactly equal durations by “a jiffie“?
Or how, specificly, do you propose that participants (such as “Al” and “Em“) who are separated from each other and who are (in general) not even at rest to each other should go about comparing what one means by “a jiffie“, and what the other means by “a jiffie“?

p.s.
I appreaciate that you seem to be similarly enthusiastic about this topic as I am; and surely you realize that I could respond to your comment more comprehensively. Perhaps you could suggest some specific example(s) of what you consider “experiments in Galilean invariance” and/or “tests of SR“, and I’d be happy to address such specific claims in a blog post (more thoroughly than seems appropriate in a mere comment).

• Wyrd Smythe

“…(supposed) physics…”

Supposed? Study any physics text book. Or see the Wikipedia article that I’ve repeatedly linked to in this series of posts.

“Do you claim that everyone, outright, means exactly equal durations by ‘a jiffie’?”

Of course not. I used “jiffie” as a fun stand-in for any concrete time unit we care to use.

I this post I wrote: “To the right is a stop watch that has measured three ticks (let’s call them milli-jiffies) for the ball to fall.”

In the post before this I wrote: “We might measure that speed in miles per hour, or meters per second, or even cubits per nano-jiffy (or one of my personal favorites: furlongs per fortnight),…”

The point is that we’re talking abstractly so the names of the units involved really don’t matter. As I’ve said repeatedly, we take it as given that Al and Em agree on them and are capable of synchronizing their time pieces.

• user12262

Wyrd Smythe wrote (March 24th, 2015 at 5:13 pm):
> […] we take it as given that Al and Em agree on […] concrete time unit[s]

We“?? No!, you’re excluding those (physicists) who would routinely ask and try to measure whether Al and Em agreed in this respect. For instance, if Al and Em had likewise been referring to their respective “time unit” as “second”, then trying to determine whether Al‘s and Em‘s samples of Cs133 atoms had indeed been undisturbed, or at least equally disturbed, or who they were different from each other (especially while Al and Em had been separate frome each other).

You’re excluding those (physicists) who appreciate and use SR (or GR) for this purpose; and who consequently would never claim having “tested SR (or GR)” but (“merely”, though thoroughly) whether given “setups” (e.g. samples of atoms, “resonators” and somesuch) did correspond to some particular model, or how to quantify any deviations.

> and are capable of synchronizing their time pieces.

Do you claim that Al and Em could succeed in “ synchronizing their time pieces” while they were moving wrt. each other??

> Study any physics text book. Or see the Wikipedia article that I’ve repeatedly linked to in this series of posts. […]

This response appears as shallow as typical run-off-the-mill physics text books, and Wikipedia articles on this topic.

• Wyrd Smythe

And your responses indicate you’re either trolling me or don’t really understand this material as well as you think you do. A lot of what you’ve written in these comments comes across as incoherent.

Scientists routinely agree on time units. The first link you included illustrates this:

Considering that a very precise definition of the unit of time is indispensable for science and technology, the 13th CGPM (1967/68, Resolution 1) replaced the definition of the second by the following:

And it then gives the precise definition scientists all over the world use for one second.

The second link has nothing to do with this, and the paragraph containing that link is particularly incoherent.

In any event, you’ve exceeded my patience, and we’re done. You’re no longer welcome here, and I will likely delete further comments.

• user12262

Wyrd Smythe wrote (March 27th, 2015 at 1:42 am):
> Scientists routinely agree on time units.

Sure they do.
Scientists routinely employ the methods of SR, and of GR, to test whether they agree on their separate realizations of “time units“, or to determine how those are related to each other. (Instead of the inverse: wanting to “test (the methods of) SR, GR” by idiosynchratic means.)

> [… The BIPM provides] the precise definition scientists all over the world use for one second.

The BIPM definition refers explicitly “to a caesium atom at rest at a temperature of 0 K […] unperturbed by black body radiation” (and, arguably, “plainly unperturbed“). However, the BIPM definition doesn’t spell out how to relate this to any concrete “hardware” given to various scientists, in various trials. Consequently: the indispensible role of SR, GR for achieving comparisons.

> In any event, you’ve exceeded my patience, and we’re done.

In any event, as long as you publicly claim that SR could be tested, or should be tested (instead of being comprehensible and applicable from the outset), you can count on being rebutted; whether you tolerate that in the form of comments, or not.