airplane on a treadmill

Okay, let’s do this. The web tubes have been buzzing about this little brain teaser for years. The even did a bit about it on Mythbusters. If you put an airplane on a treadmill, can it take off? Well, it turns out there are several different answers depending on precisely how the question is worded. But, IMHO, the vast majority of these answers have missed the point entirely.

You might think it’s a puzzle designed to see if the student knows how airplanes work. Something like this:

TEACHER: Imagine an airplane which accelerates down a mile-long runway until it reaches takeoff speed, which is 100 mph. What would happen if you put the airplane on a treadmill instead of using the runway?

STUDENT: Is the treadmill capable of turning faster than 100 mph?

TEACHER: Yes. It can turn any speed you want.

STUDENT: If you turn on the airplane’s engines and spin up the treadmill at the same time, when it reaches 100 mph the plane should take off, right?

TEACHER: (smugly) Ah, you failed to realize that the speed of the wheels relative to the surface is totally irrelevant. Airplanes do not push against the ground with their wheels the way cars do. They create lift because of their wings rushing through the air. It doesn’t matter how fast the wheels are turning. If there’s no air flowing over the wings, there’s no lift, and the airplane can’t take off.

STUDENT: (humbly) oh great teacher, thank you for explaining the difference between how cars work and how airplanes work.

But then along comes a smart ass.

SMART ASS: But wait! If the treadmill is turning 100 mph and the plane’s wheels are also turning 100 mph and the plane is sitting still, why can’t the pilot simply throttle up the engines a bit more until the plane starts to move down the treadmill? And then when the plane’s wings are moving 100 mph relative to the air, the plane will take off! As you said, it doesn’t matter how fast the wheels are turning.

TEACHER: (flustered) Well, that won’t work because the treadmill operator can just speed up the treadmill at the same time the plane is throttling up its engines. The treadmill operator can prevent the plane from moving.

Did you notice that the question has changed? A minute ago, we wanted to know if it’s possible to make the airplane take off. Now, we are discussing whether it’s possible to prevent the airplane from taking off. Those are different questions! But our smart ass isn’t done yet…

SMART ASS: How can the treadmill prevent the plane from moving? The only force it can exert on the airplane is the rolling resistance of the wheels. This force is caused by friction and it does not increase as the speed increases! If the plane’s engines are capable of producing more thrust than the friction of the wheels (and they must, or else the plane was never capable of taxiing down the runway in the first place) then the airplane can move! And if it can move, it can accelerate. Once it reaches 100 mph relative to the air, it will take off, regardless of how fast the treadmill is turning at that time.

TEACHER: (utterly defeated) I bow to your superior intellect.

THE VOICE OF REASON: Excuse me for butting in, but how long would the treadmill need to be?

SMART ASS: The treadmill would need to be just as long as the runway was.

THE VOICE OF REASON: So, if we needed a 1 mile runway, now we need a 1 mile treadmill? How is that better?

SMART ASS: uh…

THE VOICE OF REASON: Why did we want to build the treadmill in the first place?

TEACHER: The runway was inconveniently long. We wanted to launch our airplane in a shorter distance, say from the roof of an apartment building. Let’s call it 150 feet.

THE VOICE OF REASON: Okay, if you take an airplane which needs a 1 mile runway to get up to 100 mph in order to take off, and you put that airplane onto a 150 foot treadmill which is capable of turning whatever speed you want, what will happen?

SMART ASS: The airplane will move down the treadmill, fall off the end and crash.

THE VOICE OF REASON: So, if your goal is to eliminate the runway, having a treadmill doesn’t help, does it?

TEACHER: (triumphant) Yes! That’s right! The treadmill is useless. If you want to launch the airplane in only 150 feet, you don’t need a treadmill; you need a catapult.

STUDENT: I think this explains why they don’t build aircraft carriers with treadmills on them.

very sad news from Switzerland

Voters in Switzerland recently rejected a referendum which would have capped CEO salaries at 12 times the pay of the company’s lowest-paid employee. This is sad because supporters of the initiative point out that the ratio has climbed from 6:1 in 1984 to 56:1 in 2007. But what’s really really sad is that when the Swiss looked at their fat cat one-percenters making 56 times what the janitor makes they got ANGRY and said this has gone too far, gathered signatures on petitions to bring the ratio back down to 12:1… meanwhile in the USA the ratio was already above 58:1 in 1989 and now it’s up to 273:1, but where is our ballot initiative?

The Swiss get mad because 56 is too high… but here on the other side of the Atlantic Ocean we look at 273 and say “eh.. what can you do?”, shrug, and go back to watching television.

I’m a business owner. I think I can reasonably make a case that I deserve to be paid twice what the lowest-paid employee makes, or maybe three times. If you paid me four times as much, I’d feel guilty about it. Twelve times is stupid. Fifty-six is absurd. Two hundred seventy-three is insane.

It would be quite easy to fix this. All we need is a change in the tax law. Employee salaries are considered “expenses” from the company’s point of view, and you don’t pay taxes on expenses. They are deductible. Let’s cap the deduction at 12 times the salary of the lowest paid worker. If your company pays the janitor $25,000 per year, the company can deduct all of that as an expense, and it doesn’t count as the company’s profit, so they don’t pay income tax on it. Now, 12 x 25,000 = 300,000 so your company can pay the CEO $300,000 per year and deduct all of that as an expense too, no problem. What’s that you say? The board of directors wants to pay the CEO $1.3 million? They are free to make that choice, but only the first 300,000 is deductible. That extra million is counted as profit and the company has to pay income tax on it. Suddenly the board will think twice about giving the CEO such a huge raise. And suddenly the CEO has an incentive to raise the janitor’s wages. Let’s get that petition started!

But you won’t hear any serious discussion about this idea, not in the corporate media. America follows The Golden Rule – whoever has the gold makes the rules.

What’s the point of voting?

Seriously, what is the goal we are trying to accomplish with voting?

When two people are discussing an idea, they don’t bother to vote on it. They each say what they think and then come up with a plan which they both can agree on. If they can’t reach a compromise, they might decide to take turns or they might decide to split up and each do their own thing. The only reason you need to vote on something is when the groups is too large for everyone to listen carefully to everyone else’s opinions.

Okay, so when you have more than just two or three people, maybe voting helps you reach a decision faster. But there are several different kinds of voting.

Imagine a group of 7 people who are voting on where to go for lunch.

3 of them love Chinese food and think pizza is almost as good but they hate burgers.
3 of them love burgers and think pizza is almost as good but they hate Chinese food.
1 of them loves pizza and thinks Chinese food is okay but doesn’t like burgers.

In the kind of voting we Americans are all familiar with, each person gets to state their favorite and whichever favorite gets 4 or more votes out of 7 is the winner. This method is called “first past the post” or FPTP for short. Let’s see what happens when we try this method. None of the choices have 4 votes, so if everyone votes sincerely there will be no winner. Somebody will have to change their vote. Naturally, the one who wants to vote for P will feel pressured to change their vote to either C or B. If we repeat this experiment day after day, we’ll find that most of the time we end up with 4 votes for C and 3 for B but sometimes we end up with 4 votes for B and 3 for C. P never has a chance. Never.

Unless the Chinese place is closed. Then it comes down to P vs B and P wins 4 to 3.
Or if the burger place is close. Then it comes down to P vs C and P wins 4 to 3.

Isn’t that interesting? P wins against B with sincere voting and P also wins against C (it’s call the Condorcet Winner) but when you put P up against B and C together, it never wins!

Let’s think about how happy the people are when C wins. The 3 who love C are very happy, of course. And the 1 who thinks C is okay is not quite so happy. And the 3 who hate C and very unhappy. They would be tempted to bribe the 1 P voter to swing the other way next time around. But all that would do is make 3 other people very unhappy. Either way, you have 3 out of 7 being very unhappy.

What if we went with P even though it didn’t win the votes? The 3 who love C are happy with P (not as happy as they would be with C, but still happy). Same goes for the 3 who love B; they’d be happy with P. And the one voter who loves P will be very happy. P makes every single person happy! So why can’t we find a way to vote that gives P a chance to win?

One suggested alternative to FPTP is Instant Runoff Voting, or IRV. The way IRV works is that you are asked to rank what is your first choice, then second choice, et cetera. Then we eliminate the choice which had the fewest first-place votes and count what’s left.
With FPTP we got this…
3 C (sincere)
3 B (sincere)
1 C (reluctant)
but with IRV we instead get this…
3 C>P>B (sincere)
3 B>P>C (sincere)
1 P>C>B (sincere)
and then P is eliminated because it got the fewest first-place votes, so now we read the ballots as:
3 C>B
3 B>C
1 C>B
And that means C wins, again. The only thing that changed is that our 1 P voter had the opportunity to sincerely express their opinion, but then we ignored their opinion and ended up with the same result we had before.

Another alternative is Borda voting. We again ask each person to rank the choices 1st 2nd 3rd and then we assign a point values. For each 1st place vote, that choice gets 2 points. For each 2nd place vote, that choice gets 1 point. 3rd place votes get 0 points.
So, with these ballots…
3 C>P>B
3 B>P>C
1 P>C>B
C gets 2 points 3 times and 1 point 1 time, total of 7 points
and B gets 2 points 3 times, total of 6 points
and P gets 1 point 6 times and 2 points 1 times, total of 8 points.
So P wins with 8 points.
BTW, if the number of choices (N) is more than 3, the actual formula is that a P-place votes gets (N-P) points.

But did we really gather accurate information about people’s feelings? We only asked them to rank the choices, and then we assigned points. What if we allowed them to assign their own point values? That’s called Range Voting, or RV for short.
This time, let’s ask them to rate each choice on a scale of 0 to 9, with 0 being the worst and 9 being the best.
3 say C=9 P=8 B=0
3 say B=9 P=8 C=0
1 says P=9 C=5 B=4
Now let’s add up the points and see which choice has the highest average rating.
C got a total of 32 points from 7 voters; that’s an average of about 4.6 out of 9.
B got a total of 31 points from 7 voters; that’s an average of about 4.4 out of 9.
P got a total of 57 points from 7 voters; that’s an average of about 8.1 out of 9.
P wins by a landslide!

What if the ballots aren’t completely filled in?

Under FPTP, blank ballots are counted at all.

Under IRV, partially filled-in ballots may be counted. For example, if we get…
3 C>P>B
2 B>P>C
1 B
1 P>C>B
P is eliminated for having the fewest 1st place votes, which simplifies to…
3 C>B
2 B>C
1 B
1 C>B
And C wins, 4-3.

Under Borda, choices which aren’t ranked get no points.
3 C>P>B
2 B>P>C
1 B
1 P>C>B
gives 7 points for C, 6 points for B, and 7 points for P, so there’s a tie!

Under RV, blanks don’t affect the average.
3 say C=9 P=8 B=0
2 say B=9 P=8 C=0
1 says B=9 and the rest blank
1 says P=9 C=5 B=4
C got a total of 32 points from 6 voters; that’s an average of about 5.3 out of 9.
B got a total of 31 points from 7 voters; that’s an average of about 4.4 out of 9.
P got a total of 49 points from 6 voters; that’s an average of about 8.2 out of 9.
P wins.

There is one more detail about Range Voting, you aren’t allowed to win without a quorum. Suppose an 8th voter joins at the last minute and suggests Indian food for lunch. They add Indian to the ballot, but most of the other voters don’t know anything about Indian food, so they leave that option blank. Here’s what happens…
3 say C=9 P=8 B=0 I=blank
2 say B=9 P=8 C=0 I=blank
1 says B=9 I=9 P=blank C=blank
1 says P=9 C=5 B=4 I=blank
1 says I=9 P=8 B=blank C=blank
C still gets a total of 32 points from 6 voters; that’s an average of about 5.3 out of 9.
B still gets a total of 31 points from 7 voters; that’s an average of about 4.4 out of 9.
P now gets a total of 57 points from 7 voters; that’s an average of about 8.1 out of 9.
I gets a total of 18 points from only 2 voters; that’s an average of 9 out of 9 — a perfect score!
A quorum is defined as at least half the number of points as the choice that got the most points.
In this case, P got 57 points, so a quorum is 29 points or more.
I is disqualified because it doesn’t have a quorum.
P still wins.

It’s possible to win IRV without a quorum. Suppose we had eighteen people who vote like this…
6 I>C>P>B
5 C>P>B
4 P>B
3 B
B is eliminated in the first round, then P is eliminated, and finally I beats C by 6 to 5.
Two thirds of the ballots cast did not mention I at all, yet I wins.
But if you recount the same ballots using Borda, I gets 18 points, C gets 27, P gets 25, B gets 14, so C wins.
If we were using Range Voting, we might get…
6 I=9 C=8 P=7 B=0
5 C=9 P=8 B=4
4 P=9 B=1
3 B=9
so I gets 54 points, C gets 48+45=93, P gets 42+40+36=118, B gets 20+4+27=51.
Both I and B are disqualified for not having a quorum (118/2=59).
C has a rating of 93/11=8.5, P has a rating of 118/15=7.9, so C wins.

So, which of these methods is the right one? Well, there’s no easy answer to that question. It depends on how much information you really want to get from the voters. FPTP gives you the least amount of information, and it encourages voters to lie. It frequently leads you to bad decisions. If you really want good information and you’re trying to reach a decision that involves the greatest good for the greatest number, I suggest Range Voting. But it’s complicated to explain to people who have never seen it, and it takes a bit of arithmetic to figure out the winner. When computers are handy, this is not a problem. But even RV suffers from the fact that you’re voting on a tiny list of choices in a universe where the actual choices are practically unlimited. In the example above, maybe the right thing to do is to take turns.