Ant On A Rubber Rope
en.wikipedia.org
en.wikipedia.org
The expansion of the rope is over its entire length. When the ant is halfway across the rope, an expansion of 1 km only adds 0.5 km to the length the ant must traverse. So the farther the ant walks, the less the expansion matters, and eventually it won't even add the 1 cm the ant is moving per second.
At least that's as best I understand it.
Here's a Google spreadsheet that shows the process with a significantly faster ant:
https://docs.google.com/spreadsheet/ccc?key=0AsmKPGVX0X-0dFF...
No guarantee of being free of off-by-one type of errors, but it should give the general idea.
You should add it to Wikipedia, maybe under a new "An intuitive explanation" paragraph.
We can represent this insight with a series: (1+1/2+...+1/n), as the wiki article states in one of it's approaches. Using Big-O logic, we know a series of this form is a Harmonic Series, and any Harmonic Series is O(log n). This is good news as there is no upper bound and will eventually reach 1 km (...someday).
Alternatively, say the rubber-band doubled every second? The distance the ant must travel is now growing exponentially and he will never get to the other end.
Awesome explanation.
Almost every piece of information you get from this hypothetical scenario is useless. Everything you know about ants, rubber, gravity, energy etc is suddenly a distraction from the extremely narrow set of ideas which the author allows to be relevant to the problem.
As someone who really really likes knowing material properties, insect physiology, and basic physics, while caring little for abstract mathematical properties, this kind of problem seems perfectly designed to tweak my nose.
Here's the problem
"An ant starts to crawl along a taut rubber rope 1 km long at a speed of 1 cm per second (relative to the rubber it is crawling on).
At the same time, the rope starts to stretch by 1 km per second (so that after 1 second it is 2 km long, after 2 seconds it is 3 km long, etc).
Will the ant ever reach the end of the rope?"
What is so hard and frustrating about figuring this out? If you over think it, blame yourself. This has nothing to do about ants, rubber, gravity, energy, it's a simple question, solving it is not so simple. But if you think of it in a simple form, you can intuitively see that the ant can reach the end of the rope.
I don't see a way around it, but it's a valid complaint about word problems that involve unusual scenarios. The "One train leaves blah, other train leaves blah, find where they cross paths" word problem is in tune with the real world, whereas an infinitely stretchy rubber band with an ant isn't.
The effect of the stretching is to reduce the ant's speed measured in units of "proportion of the rope". E.g. if the rope is 100cm and the ant is going 1 cm/s, then the ant is covering 1% of the rope per second. If the rope doubles in length and the ant keeps the same speed, the ant is covering 0.5% of the rope per second. So the ant is decelerating in this set of units; the total distance it travels becomes a matter of computing a limit.
Correct result, but that reasoning is dodgy.
It's easy to imagine a different problem where an ant always covers a positive proportion of the rope every unit time, but the ant doesn't reach the end of the rope. E.g. construct a situation where, in the 1st unit of time, the ant covers a quarter of the rope, in the 2nd an eighth, in the 3rd a sixteenth, ..., in the nth a 1/2^{n+1}th. In that scenario, as time tends to infinity the ant asymptotically reaches the half-way point.
The reason it works in the problem in the article is that, there, the additional proportions the ant covers in each unit of time form a divergent series (~the harmonic series), meaning the sum tends to infinity as n-> infinity. In my version above, they form a convergent series, meaning the sum tends to a finite number (in my example, 0.5) as n-> infinity. (Of course, a convergent series with a sum >1 would also mean the ant reaches the end of the rope in finite time).
.00001/1 + .00001/2 + .00001/3 ... => infinity. The ant will eventually pass 100% of the band.
There are lots of counter intuitive results based on the fact the harmonic series has no bound. I remember an article where someone showed you could stack dominoes to produce infinite overhang, exploiting the same counter intuitive bit of math.
Does this make sense?
Edit: added the question sign.
An ant starts at one end of the rubber band (POINT A), and walks toward the other end (POINT B), at 1 centimeter per second, for an infinite period of time.
POINT B moves away from POINT A at 100,000 centimeters per second, for an infinite period of time.
POINT B travels at: 100,000 centimeters / second
The ant travels at: 1 centimeter / second
In 100,000 seconds, the ant reaches the 100,000
centimeter mark, but POINT B is now 10,000,000,000
centimeters away. Both continue to move at the same
speed.
If an average ant lives 90 days, then the ant only has 7,776,000 seconds to reach the end of the rubber band. In this time, the ant will travel 7,776,000 centimeters, but the rubber band will have extended to 100,000^7,776,000 (that's one hundred thousand to the roughly seven-point-seven-millionth power) centimeters long. Based on these facts, we may safely conclude that a mortal ant will never reach the end of this hypothetical rubber band.
Did I miss a detail somewhere?
(Also: this is not a problem concerned with the physics of rubber or biology of ants. This ant is immortal, the rubber infinitely stretchy, etc.)
Which is great, except that when the original concept is framed to the reader, the simple word problem doesn't specifically express that both ends are in motion.
If both ends are in motion, then the ant's velocity changes. The ant might move 1 centimeter per second, relative to a fixed point in space, but it isn't moving 1 centimeter per second, relative to it's origin, POINT A.
The motion of the rubber band is influencing the speed of the ant, relative to the band itself. Since the ant moves toward POINT B, while POINT A simultaneously accelerates away from the ant, the ant is moving faster than 1 centimeter per second, with each second spent traveling away from POINT A.
Once this detail is added, it completely changes the premise of the problem. But the manner in which the problem is originally expressed is vague and incomplete, misleading the reader (perhaps deliberately so).
Why would anyone claim the ant is moving 1 centimeter per second, RELATIVE TO THE ONLY OTHER FRAME OF REFERENCE IN THE PROBLEM (the rubber band), when its motion is clearly not 1 centimeter per second?
In that case, the ant is only moving at a fixed speed, toward the other end (100,000 times slower than the end's expansion), which will forever be too far away.
It became one of my favorite puzzles, and I tried it on friends and family. I certainly never thought it could have "real-world" applications.
But then it occurred to me, people were solving this puzzle every day.
Think of an ordinary loan. Each month, you make a payment that's part interest and part principal. At first it's mostly interest, then the ratio shifts over time, the final payments being nearly all principal.
You normally start with a loan amount (principal), an interest rate, and a loan duration (typically 180 or 360 months). From these parameters, you figure the monthly payment, a process called amortization. Part of the payment goes towards interest, part towards the principal.
Each month you're required to service the loan, which means to pay (at least) the interest. That's the cost of renting the money. Some loan arrangements allow you to pay down the principal at your own pace. If you only ever pay interest, the principal will remain unchanged, and the loan will go on forever. This is the Netflix model: they don't care how long you keep a disc—you're paying rent on it every month. Many people pay more than the monthly payment from time to time. The principal will be reduced by this extra amount.
The worm has taken out a loan. The twist is, we don't yet know the principal, nor the total amount to be paid, which corresponds to the final rope length. Instead, we know the payment: 1 yard. The interest portion varies, but the worm consistently pays down the principal 1 inch each pay period.
Some of the added yard (monthly payment) appears in front of the worm (interest portion) some behind it (principal portion). Stretching the rope uniformly has the effect of servicing the interest. At first, most of the newly added rope appears in front of the worm, but as with the ordinary loan, the back/front ratio increases over time.
That the worm will eventually reach the end of the rope is now evident. If your interest payment is taken care of, then even a small monthly pay-down of the principal will eventually pay off any size loan.
The really interesting problem is an ant walking on the surface of an expanding sphere, where the rate of the expansion of the sphere is proportional to the radius of the sphere.
If the ant starts at one pole, will it ever reach the other pole?
Yes, I've underspecified the problem - we want the complete set of solutions.
Assume initial radius r, ant of length l, ant walking at speed w, and initial rate of expansion e (remembering that de/dt is proportional to r).
Under what conditions will an ant at pole P1 reach P2? Under what conditions is impossible for the ant to reach P2?
Presumably there is a range of values such that the ant can traverse the sphere if it starts early enough or is big enough or if w is high enough.
Now make things even more interesting: Assume that there is a maximum speed M, and that the closer w or e gets to M, the harder it is to get incrementally close to M.
This is the expanding universe. We are the ant.
If we had developed high speed interstellar travel "early enough", would we ever have been able to cross the universe? Or is it always too late?
Now that's an interesting problem. Solution left as an exercise for the reader.
After one step, the ant will be somewhere in the middle, and both ends will be moving away from it. (Because when the rope stretches, every point on the rope moves away from every other point on the rope.) If the far end is moving one direction, and the near end is moving the other direction, and the two ends are moving away from each other at 1 km/s, then neither of them can be moving away from the ant at 1 km/s. The ant will then be able to catch up with either end.
Details matter.
Other fun questions: does the rope stretch uniformly or are there waves? Do the ants fore and rear legs get separated as the rope under it stretches? Is there any sag in the rope?