### Speed Math

Over the course of my childhood, I remember asking exactly one intelligent question. Unfortunately, I couldn’t make my parents understand what I was asking. Perhaps it was that frustration that deterred me from ever formulating an intelligent question again.

I was, I think, six years old at the time, and my question was this: If you’re traveling at 50 miles an hour at 1:00, and you’re traveling at 70 miles an hour at 2:00, must there be a time in between when you’re traveling exactly 60 miles an hour?

What made this question intelligent—and probably what made it incomprehensible to my parents—was that I was very keen to distinguish it from the question of whether your speedometer would have to pass through the 60-mile-an-hour mark. It seemed clear to me that the answer to that one was yes—that even if your true velocity could somehow skip directly from 50 to 70, the speedometer needle, in the course of whipping around from one reading to the other, would have to pass through the midpoint.

In retrospect, though, what I was trying to get at was the distinction between the Intermediate Value Theorem, which applies to continuous functions (like your speedometer reading) and Darboux’s Theorem, which applies to derivatives (like your velocity). And I had exactly the right intuition, which is that the Intermediate Value Theorem is easy but Darboux’s Theorem is (comparatively) difficult. In other words, it’s pretty much obvious that the speedometer has to pass through 60 but not so obvious that your actual speed has to pass through 60, although in fact it’s true.

Perhaps a much more intelligent (or obstreperous) child would have questioned the continuity of the speedometer reading, or the continuity of space and/or time itself.

What is the most intelligent question you can remember asking as a child? Did you get a good answer?

#### 35 Responses to “Speed Math”

1. 1 1 Bennett Haselton

Minor nit-pick: According to Darboux’s theorem, if your speed is 50 at one time and 70 and a later time, then your speed has to be 60 at some point in between, *provided* that your speed is *defined* at all points in between. (In other words, that your graph of distance traveled versus time, is differentiable at all points on that interval.)

It’s logically conceivable that an object could start moving at 50 mph, and then at time t=1 it starts moving at 70 mph, and there is never a time at which its speed is 60 mph — however, at time t=1, its speed is undefined. If its speed is well-defined at all points then at some point it has to pass through 60.

Side note: What’s the simplest function you can think of that is differentiable everywhere, but not *continuously* differentiable? I started at this for a long time on a plane flight, thinking that all differentiable functions had to be continuously differentiable. (Which, if that were true, would make Darboux’s Theorem trivial, by the intermediate value theorem.) But then I found a counterexample:

f(x) =
0 if x = 0
(x^2)*sin(1/x) if x != 0

Although there may be simpler ones.

2. 2 2 marko

I remember when I was maybe 10, we had a class called Nature, or something like that. Teacher talked about the Solar system and how it was created from gas and dust spinning around. Later she talked about planets and how different they are. I remember asking her “But if all planets are made from the same gas and dust, how can they be so different?” She said she didn’t know…

The thing is that I never thought of that again and I still don’t know the answer. I’ll google it now….

3. 3 3 thedifferentphil

While it might have seemed obvious to all that the needle would have to go through the 60mph line, sometimes “obvious” is wrong, and asking the obvious questions is how you sort out the world. As you point out in your book that electrons do not move along curves around a proton, but pop up here and there noncontinuously.

4. 4 4 Alan Gunn

I’m pretty sure I never asked an intelligent question as a child, though I like to think I’ve come up with a few as an adult. I remember childhood as a time of soaking things up, but pretty passively. Sort of a read every book you can find approach. I can also remember asking an adult family member a particular question (about how the clutch on a car worked) and getting a response that clearly showed that he didn’t know and wasn’t going to admit it. I think it was that experience that left me with a strong dislike of bull\$hit. I have since been surprised by the number of people who just don’t mind being bull\$hitted, who even seem to enjoy it.

5. 5 5 Thomas Purzycki

While I can’t recall asking anything intelligent in the same sense as the velocity question, I look back with fondness at the questions I asked expressing my frustration at primitive technology of the late 80′s/early 90′s: Why can’t I watch TV shows on the computer? What do you mean we don’t know for sure if there are planets around other stars? Why can’t people go into suspended animation (after seeing “Alien”)? Why doesn’t someone bring back the dinosaurs for real (after seeing “Jurassic Park”)? Luckily, many of these frustrations have been addressed and I have high hopes for some of the others.

6. 6 6 Rowan

I don’t remember it, but my mother tells me that when I was five I marched up and demanded to know if there was time before time. She also tells me that her answer — that there were many different ideas about that — was thoroughly unsatisfactory.

7. 7 7 Dave

When I was starting to understand numbers and elections, I remember asking my father why he bothered voting because it seemed like a waste of time.

His response was something along the lines of “if everyone thought that way” which I found amazingly profound.

8. 8 8 Thomas Bayes

This reminds me of a simple — but fun — problem. A car is traveling with a velocity of 50 miles per hour for 100 miles. What does its velocity need to be for the next 100 miles so that its average velocity is 75 miles per hour for the entire 200 mile trip? It is even better to ask what its velocity needs to be to average 100 miles per hour for the entire 200 mile trip.

(This is a good way to get people to think about the difference between velocity and pace. Similar problems can be constructed with fuel efficiency to see the difference between using miles-per-gallon and gallons-per-mile, which is critical to get people to make wise choices about improvements in fuel efficiency. Is it, for instance, better to improve a truck’s gas mileage from 12 mpg to 14 mpg, or to improve a car’s from 25 mpg to 35 mpg?)

9. 9 9 Required

When I was about 7 years old, my dad was telling me something about the revolution of the earth around the sun and the moon around the earth and eclipses and so on. I assumed all this is taking place in one plane, and asked why we don’t have solar eclipses approximately every month.

10. 10 10 Al V.

I don’t know if this counts, since I was in high school when I asked the question. When we learned (at a cursory level) about the special theory of relativity, I wanted to know how we could be sure that the speed of light had always been 300M m/s? It seemed to me (and still does) that at lot of our explanations would be radically different if the speed of light was different billions of years ago. Nobody has ever explained to me how we know that it has remained constant.

11. 11 11 nobody.really

1. “How many sides does a point have?” (Answer, after long digression into the meaning of “side”: one.)

2. “Why is that guy [a water skier] pushing that boat around the lake?” (Never did get an answer; everyone was laughing.)

12. 12 12 Neal Davis

I don’t recall asking about it, but I do recall coming up with the idea of a derivative: could one know the rate of change of something else? It was several years later when learning about the derivative that I made the connexion and instantly understood that portion of the calculus conceptually.

13. 13 13 miko

“Is the yellow i see the same as the yellow you see?”
I recall being frustrated by the lack of good response to this question in elementary school.

14. 14 14 improbable

I asked the question about whether we all see the same colours, too. I remember thinking that this was deep, although now I think it shallow. This is why I’m not a philosopher…

Your question about the car’s speed marks you as being on the mathematician side of the physics/maths divide. As a physical matter it is no less obvious that the car’s speed must pass through 60 as that the speedometer’s needle must do so. The motion of big floppy steel objects like cars is smooth to more derivatives than needed.

15. 15 15 Super-Fly

I remember in 2nd grade or so when I figured out that my ‘mind’ was completely distinct from that of my peers. Meaning, there was no way I could ever ‘see’ the world through Laura’s (one of my friends) mind. I was pretty sure that they had thoughts and feelings like I did, but I could never confirm that fact. I suppose this is a rudimentary musing on mind-body distinctions.

I also figured out that the area of a triangle was 1/2 base*height on a quiz in 4th grade. It’s too bad my rate of intelligent questions has dropped so remarkably.

16. 16 16 Harold

Thomas Bayes: – yes, quite fun that one. I make it 150 and infinity. It is a bit counter-intuitive. A small technical point, would it not be speed, as a scalar quantity? Velocity includes direction. You could drive round in a giant 200 mile circle at an average speed of 75 miles an hour, but average velocity is zero (I think).

I couldn’t figure out rainbows for a long while when I was a kid (and probably an adult). Seems so obvious now, I wonder what part I found difficult.

17. 17 17 Todd

When I found out in elementary science that light reaches the Earth in about 8 minutes (teacher’s words), a light bulb went off. I raised my hand to wonder aloud why it mattered to study the sun since everything we saw was no longer true; we were actually observing the past. Then I postulated the star formations were also not up to date. So I asked why we had to study constellations when they aren’t real? Of course today I understand the value of such study, but it seemed the adults (and certainly not my classmates) never grasped my point: Looking at distant objects is observing their past, not their present. I wanted to see what they looked like in their “now.”

18. 18 18 Mark

I’d still like good explanations to these: (would either of them make good blog posts?)

1. Is there an upper limit to how hot something can be? (After hearing that the speed of light was the fastest something could go and that something gets hotter when the molecules move faster). I asked every teacher I had for years and never got a good answer. (Years later I found the answer to this on Google, but it wasn’t a good explanation for a non-science person)

2. (I asked this in college so I don’t know if it counts) Could a Foucault pendulum be used to find an absolute reference point? On earth it seems to be the case (put me in a train with no windows and vibrations and I can still tell if I’m ‘moving’ by using a Foucault pendulum). The teacher suggested we move the experiment to space to get rid of the effect. However, couldn’t the two astronauts who fly by each other (each thinking they’re staying still and the other is moving) resolve the dispute by using the pendulum (because the solar system is rotating)? If we move the experiment out of the solar system, would we still find the effect at the galaxy level? And if we left the galaxy, would we find the same effect at the universe level? If so, is that an absolute reference point?

19. 19 19 Seth

One memory from my preschool days was me tearing a piece of paper and thinking that if I could keep tearing it in half, it would get so small I couldn’t see it. After making several tears, I asked the teacher why that didn’t work and she gave me the “you-must-be-quite-the-odd-duck” look.

When I was older, I took a car trip to Mexico. I crossed from the U.S. to Mexico at Laredo, TX. I wondered why folks on one side of the river had a much better standard of living than folks on the other side (that thought may have been helped along by the fact that I live in a town where I cross a river daily and don’t notice apparent differences in the standards of living on either side). I kept wondering about that and now I’m pretty much libertarian, I think.

My kid asked me last night why Walgreen’s was named Walgreen’s. “Is it because they have green walls?” I thought that was funny.

20. 20 20 Roger Schlafly

So your question was intelligent because you anticipated the possibility that the car’s speed was differentiable but not continuously differentiable?

If you were my kid, I would assume that you did not understand that the intermediate value theorem could be applied to the derivative.

21. 21 21 Neil

I stumped my high school physics teacher, and felt the question was smart at the time. The teacher was explaining the refraction of light and how light bent as it went through glass because it travelled more slowly in the glass, but it “straightened out” again when it left the glass. I asked what sped the light back up again when it left the glass, and the teacher was stumped and rambled on incoherently for 15 minutes.

22. 22 22 improbable

“Is there an upper limit to how hot something can be?”

There’s an upper limit on the temperature at which we still have any confidence of our understanding of physics. In fact it’s a bit stronger than that, above the Plank temperature we know for sure that we are ignorant of how things work.

When we look at the black sky, we are looking at the very early universe, which was very hot, about 3000K at the time of becoming transparent. And the further back you go, the hotter it was. The simplest model for this has a big bang, an actual moment t=0 at which the temperature diverges. But we are sure that we cannot trust such a model all the way back to this time.

I guess that doesn’t really answer your question, so maybe we just don’t know the answer, yet.

23. 23 23 Ken B

How do you know pi is the same for every circle?

No, I did not get a good answer. I did in university, but alas I now forget it.

24. 24 24 improbable

Ken B: “How do you know pi is the same for every circle?”

Here’s one argument: suppose you use as your ruler the line from the centre of the circle to the edge. The length of the circumference is Pi units of this ruler. Since there are no other units available, this number can’t depend on how big the circle is — big compared to what?

If you draw circles on a curved surface, then of course this ratio isn’t fixed. But the radius of curvature gives you a number against which to measure the radius of the circle, breaking my argument.

25. 25 25 Benkyou Burito

It was a good question, because seemingly analogous questions can have a different outcome. Your lit electric lightbulb receives 110v when the switch is closed (state 1)and 0 volts when the switch is open (state 2). It takes 1/15 seconds for the lightbulb to transition from illuminated to dark. At some point during that process is the lightbulb receiving 55v?

26. 26 26 Benkyou Burito

shoot I forgot my own question. Sorry.

When I was 8 yrs old I asked why they didn’t make slices of peanut butter in the same way they made slices of cheese. When I was 14 I filed a provisional patent application for just such a thing. Within 4 weeks my invention was challenged and an offer to purchase my rights to it were made by the same company. I took the deal and bought a PWC.

I guess someone came out with them a few years back but I have no idea if its the same group that bought my idea.

27. 27 27 Pete

There is no oxygen in space since it is a vacuum. Regardless of temperature, how does the hydrogen in the sun burn (“burn” being the word the science teacher used)?

Fortunately for me, my parents knew, but looking back on it, I’m sort of disappointed in the fact that my sixth grade teacher was clueless.

28. 28 28 dave

its never too late, eh?

if i had two giant rings of conductive material (one a ‘u’ shaped ring and the other inside it) bolted down (the u ring is bolted, the inner ring is allowed to slip) at the south magnetic pole, could i use the rotation of the planet and its own magnetic field to generate energy?

29. 29 29 nobody.really

Could a Foucault pendulum be used to find an absolute reference point?

Oooh, I like that one. I now recall a high school query: Is the concept of electrical “neutral” an absolute or a relative concept?

For example, imagine if a jillion electrons fell on earth such that the background level of electrical charge (ground) of today had a negative charge relative to the ground of yesterday. I’d guess that electricity as we know it would continue to function – that is, electrons would continue to be attracted toward places of relatively positive charge and away from places of relatively negative charge.

But consider the consequences at an atomic level: The nucleus of atoms tends to have a positive charge. In a sea of additional electrons, I’d expect each nucleus might attract more electrons than was previously the case. Atomic mass would increase. Valances would be disrupted. Would traditional bonds continue to hold? Would atoms in familiar molecules continue to “fit”?

30. 30 30 nobody.really

Continuing on the absolute vs. relative theme, I used to ask whether the Western tonal structure is “natural.”

The West has developed an eight-tone octave (Do, Re, Mi…). Other societies have developed other tonal schemes. That fact strongly suggests that the Western one is merely cultural.

Yet the Western scheme also reflects the overtones of a vibrating string. The overtone series is not merely the artifact of Western culture. A string will vibrate the way a string will vibrate, even in Jakarta! This suggests that the Western tonal structure has a status that does not derive simply from culture.

After repeated inquiry, I’ve amassed the following answer: Yes, the Western tonal structure reflects the vibrations of a string (more or less; the emergence of “tempered” instruments such as the piano has led to a bit of fudging, but oh well….) But the West places an unusual emphasis on stringed instruments. The overtones structures of vibrating wood/drum heads/metal bars/metal tubes/metal bells can be entirely different, and other cultures place greater emphasis on such instruments. And the human voice seems to be capable of producing a variety of overtone patterns.

In short, the overtones of a string are not merely a product of culture. The choice to base a tonal system on those overtones IS a product of culture.

31. 31 31 nobody.really

How do you know pi is the same for every circle?

It isn’t.

32. 32 32 GregS

I remember as a young child imagining bringing my finger closer and closer to some object (for some reason my minds eye chose a dandelion as the object), and asking, “How close can I get until I’m just barely touching it?” It seemed to me that I could always bring my finger a little bit closer, but I was convinced that there was a “limit” to this. I don’t remember asking an adult, but I probably asked my much smarter older brother.

I also remember knowing that the hour, minute, and second were all units of time displayed on a clock and represented by the movements of clock hands. I noticed that some second-hands moved continuously and some jerked forward every second. I insisted that the unit of time for the continuous hand was a “jiffy” (this is also something I remember sharing with my older brother, and he took exception to my terminology).

Much later, I remembered these things and realized that my young self had invented infinitesimal length and infinitesimal time, “dx” and “dt”.

33. 33 33 Sam Schulman

In 6th grade, I had to read a short story by John O’Hara which was extremely allusive – I really couldn’t understand a word of it (you had to know about the Porcellian Club at Harvard, about the America First Committee, about McCarthyism – how the story got into a middle-school reader I don’t know).
I got my mother to explain it to me, and I asked her (obviously in response to something she said in passing), “Mom, where can I find out all the things that everybody knows?”

34. 34 34 Michael

When I was in fourth or fifth grade, my grandfather (who was a ham radio operator) showed me air and ferrite coils used in radio, and gave a simple explanation of how they worked. I knew what the magnetic field looked like through such a coil, but then I wondered what it would look like if that coil were wrapped around another coil, then what would happen if a wire were wrapped around another wire and then wrapped around a coil with a different current flowing through it. The I wondered what a the field would look like in a coil wrapped around a coil wrapped around a coil wrapped around…ad infinitum.

I still don’t know the limiting case, but it probably looks very cool. :)

35. 35 35 vivek

‘In other words, it’s pretty much obvious that the speedometer has to pass through 60 but not so obvious that your actual speed has to pass through 60, although in fact it’s true.’
An interesting question for the future mathematical economist to ponder at such a young age.
Of course, it must be said, the speedometer doesn’t measure speed. Imagine a car chase in San Francisco, presumably the speedometer goes haywire when the car, after cresting the hill top, is flying through the air- the actual velocity at this point is not measured though we can conjecture what it might have been after getting a reading further down the road.
If there were an ‘escalator road’- on the principle of the moving walk-ways at airports- then the velocity could change discontinuously.
This raises the question for Economists, what sort of mistakes do we make when we proceed from an intuition of the speedometer sort to other metrics? Paths in the sorts of spaces Economists are interested in, will be snakes and ladders type paths.