Comments

I had trouble with the moon-earth problem....though I did get the elephant problem, both conceptually & mathematically.

One thing I've been realizing is that I need to learn Dan's dimensional analysis dominoes, and do more work with proportion & ratio. (Russian Math is brilliant on this subject.)

For me, expressions like "16 times the surface area" mess up my ability to make comparisons.

I could understand the elephant, because 'twice the size' did translate to 'twice as tall,' and I was able to make the jump that the answer would be 2 x 2 x 2 (though I wouldn't have made that jump without the multiple choice answers).

When you're talking about spheres & saying '16 times the surface area' I lose sight of the math & the concept, not to mention the unit of analysis. (I have to re-take geometry in a serious way, too...)

second thought: I know what's throwing me off. The elephant story involves height: the adult is twice as tall as the adolescent. The earth/moon comparison involves the surface area of a sphere, and I'm not instantly seeing that the surface area of a sphere translates to its 'height' or diameter. This is a case where the language is unhelpful; if he had said that the earth was 64 times 'as big' I probably would have made the conection.

This is something Anne has talked about, I think: the difficulty of translating mathematical conepts into words. You introduce extraneous notions & confusion every time you have to use words, it seems.

With the cars, I'm not even going to try....BUT based on real life, I think the big car is going to roll over the little car. That's why it's so deadly to be in the small car. If it just moved backwards, the passengers would be in better shape.

We just had a horrific accident here in New York state where some huge truck hit, head-on, a tiny little sub-compact carrying lots of teens from a local camp.

All of the car's passengers died. They showed the photographs, and there was nothing left of that car. The truck rolled over it.

So....based entirely in life experience, I would say that the little car does come to a full stop, but the big car does not.

-- CatherineJohnson - 13 Sep 2005

Of course I instantly set out to buy the statistics book, but it's $17 for 120 pages.

If it's a brilliant book, that would be money very well spent; short is almost always better (meaning smarter).

But given the level of math writing in this country, I have my doubts.

-- CatherineJohnson - 13 Sep 2005

Speaking of good books, does anyone have a recommendation for a good calculus book? I still have mine from college, but I have a hard time understanding the proofs. It seems like there are steps missing.

In the book Countdown about the math Olympiad, the author mentions that one of the mathematicians who constructs the problems is Russian. This mathematician says that his calculus book was only 100 pages long, but it was excellent and had no extraneous information. I wonder if anyone has translated a Russian calculus book.

-- AnneDwyer - 13 Sep 2005

I remember loving the Spivak book when I was in college. The great thing about that book is that it's actually quite intimidating at first glance (it was smaller & denser than other textbooks, with an all-black cover with just the word, "Calculus" printed on it), but the writing was brilliany (and quite funny at moments). I wound up using that book more than the 'official' book my class was assigned (Spivak was used by the nerds in honors calc; the wannabe nerds in my regular calc class got stuck with the boring book).

-- IndependentGeorge - 13 Sep 2005

I suspect the reporter is actually conflating force, energy, and momentum. Momentum is the mass times the velocity; assuming for the moment that the two vehicles are travelling at the same velocity, the momentum of the truck is going to be larger than the mini. Since momentum is conserved, that means the velocity of the mini will necessarily be reversed (since they were travelling in opposite directions, they have opposite signs; the magnitude of the truck's momentum is greater than the magnitude of the mini's. After impact, the mini has to reverse direction to conserve momentum).

After impact, one of three things has to happen: (1) the two vehicles 'fuse' together as one, travelling in the truck's original direction at reduced speed (an inelastic collision), (2) the truck comes to a dead stop, while the mini reverses direction at a high velocity, or (3) both vehicles reverse direction, with the truck travelling very slowly and the mini flying backwards at extremely high speeds. Since the truck has greater momentum to begin with, and mass does not change, the mini will always experience a much greater change in velocity.

This is why the mini gets totalled - IIRC, kinetic energy is the mass times the velocity squared. The mini experiences a much higher change in velocity - and therefore an exponentially larger change in energy.

At least, that's what I think happens; it's been years since I did physics, but I think those are the relevant events.

-- IndependentGeorge - 13 Sep 2005

I just realized I left out force. Force is the mass times the acceleration - what we're talking about here is not so much force, but the change in force. I don't have time to do the calculations right now, but since the mini experiences the greater change in velocity, it also experiences the greater change in acceleration. The magnitude of the change in velocity is therefore linear with respect to the force. The mini has a lower mass, but experiences a greater change in acceleration. I think these cancel out such that the change in force is the same.

I think. I can't do the calculations right now (and I'm too rusty to really trust them, anyway), but that seems to be the right explanation.

On further reflection, I don't think this is a good example of illustrating innumeracy; it's more a physics/language problem than anything else. Force, energy, and momentum have specific meanings in physics, whereas the three terms are used interchangeably in common English.

Boy, I really need to either change jobs, or stop reading this site at work. You're kiling my productivity!

-- IndependentGeorge - 13 Sep 2005

Catherine,

While I don't know if my dimensions worksheets would be any help, you are right that dimensions add insight to these problems. You explained that you couldn't get comfortable with the surface area comparisons. I think it gets easier if you do one more layer of translation: go all the way to the units. Surface area would be in units of SQUARE feet or SQUARE meters or SQUARE miles. Think of these as ft^2, m^2 and mi^2. Height would be in ft, m, or mi. Volume would be in ft^3, m^3, or mi^3.

So, proportional objects whose heights differ by a factor of 2, would have different area by a factor of 4 (2^2), and different volume by a factor of 8 (2^3).

-- DanK - 13 Sep 2005

And another question: what exactly is the physical law behind the statement "every action has an equal and opposite reaction"?

-- CarolynJohnston - 13 Sep 2005

Well, Newton's third law of motion is:

"For every action there is an equal and opposite reaction."

So I guess that's your answer. 8-)

It's actually one of the fundamental laws of physics, and usually treated as irreducible. In that regard, it's kind of like the "law" that parallel lines don't intersect.

On the issue of which vehicle experiences the most force, the short answer is "neither". The forces are the same because each experiences only the force imparted by the other. (We'll leave out, as the author of the original problem did, any consideration of ground or air friction or other confounding issues.)

The Mini exerts some force on the UPS truck, which must exactly equal the force it experiences. This is Newton's third law in its purest form.

You can derive the velocity of either vehicle after the collision from the velocity of the other vehicle, and the relative accelerations from that and the time required for the collision. The acceleration is (change of velocity)/(time to complete the collision), which will be much greater for the Mini than for the UPS truck. But the forces are the same.

As an aside, the (time to complete the collision) term shows why crumple zones reduce the danger of collisions to vehicle occupants. The longer the collision takes (the more time the vehicle crumples before bouncing, basically), the lower the peak acceleration and the less the damage to the occupants.

-- DougSundseth - 13 Sep 2005

Newton's Third Law is now viewed as a primitive version of the Law of Conservation of Momentum.

-- BernieJohnston - 13 Sep 2005

You can derive the velocity of either vehicle after the collision from the velocity of the other vehicle, and the relative accelerations from that and the time required for the collision. The acceleration is (change of velocity)/(time to complete the collision), which will be much greater for the Mini than for the UPS truck. But the forces are the same.

So my off-the-cuff derivation was right! Woohoo!

The Mini exerts some force on the UPS truck, which must exactly equal the force it experiences.

Doesn't this just say that the forces on each vehicle must equate? Is that what is meant by the phrasing of N's 3rd law in terms of 'action'?

-- CarolynJohnston - 13 Sep 2005

"Doesn't this just say that the forces on each vehicle must equate? Is that what is meant by the phrasing of N's 3rd law in terms of 'action'?"

Exactly.

-- DougSundseth - 14 Sep 2005