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bonanova · Daedalian Member

A twist on Zag's chestnut: Ant on an Elastic

The answer clearly depends on the relative speeds of the ant and car. For vanishingly small car/ant speed ratio, the ant reaches the car in 1 hour. For sufficiently large car/ant ratio, the ant will never reach the car. So the interesting question seems to be, what is the minimum ratio of car/ant speed that just keeps the ant from reaching the car? Assume as before that the ant starts walking at 0.1 mph along the connecting elastic when the car is 0.1 miles away.
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gftt* · Guest

Zag · Tired of his old title

I don't see how the ratio can make any difference, as long as it isn't infinite. Whatever the ratio is, you can make twice as many marks on the elastic and the ant will always reach one mark after the other. It might take an exceedingly long time, and heat death of the universe will happen first, but we aren't concerned with such trivialities in this puzzle.

lostdummy · Daedalian Member

Yes, I agree that for ANY speed ratio, ant can reach car. To see it based on chestnut explanation, you just need to make 2*(speed ratio) marks on elastic.

Now, more interesting question would be to show formula that calculate how much TIME ant would need, based on their speed and initial elastic length.

Because I have suspicion that time would increase exponentially - while first mark is moving away from ant at same speed as last one will, fact is that before ant reach last mark, distance between last mark and car will be much-much larger than distance between fixed point and first mark, thus ant would need much more time to cross last segment than it needed to cross first. And that thing (time to reach car) will certainly be dependent on ant/car speed ratio.

novice · No harm. Pun intended!

This sounds suspiciously like a differential equation.

gftt · Daedalian Member

Measure the ant's speed not in a fixed unit of length per time, but in length of elastic per time. Its velocity is then inversely proportional to time (v = C/t for some C, t starts at 1). You want the integral of the ant's speed to total one length of the elastic. Integrating C/t will eventually get to any total you want, but it will take exponential time.

bonanova · Daedalian Member

bonanova · Daedalian Member

Since the ant will catch a car moving at any constant speed, let's revise the question. Can the car "escape" from the ant by accelerating? If so, describe an acceleration behavior for which the car will remain ant-free.

Method: Consider the elastic is initially 1km in length, the ant crawls at a speed of 1 cm/sec, and the car initially moves 1km/sec. If we permit the car to move instantaneously 1 km at the end of each second, the ant's progress along the elastic is a (divergent) harmonic series that will eventually reach 100,000. The car will escape the ant if that series is changed to one that converges.
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