Poor snail!

Calculus Level 3

A snail starts on one end of a bungee chord that is 1 km long and can be stretched infinitely long.

Every minute he goes 1 cm along the chord.

However, unfortunately, every minute the chord also gets stretched one more kilometer. (That is, after one minute it is stretched to 2 km, after 2 minutes it is stretched to 3 km, and so on)

Will the poor snail ever reach the other end of the chord?

Assumptions: The chord stretches uniformly.

This is not an original problem, but was presented to me by my dad, Richard Pilling. :-) I understand it's a Martin Gardner original.

No Yes

1 solution

Geoff Pilling
Nov 16, 2016

We can consider the percentage of the chord the snail has traversed every minute.

In the first minute he has traversed $\frac{1}{100,000}$ of the chord. After the second minute, he has traversed an additional $\frac{1}{200,000}$ of the chord, for a total of $\frac{3}{200,000}$ of the chord.

In fact, after $n$ minutes, the percentage of the chord he has traversed is given by:

$P(n) = \frac{1}{100,000}(\frac{1}{1}+ \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n})$

And we know that as $n$ approaches $\infty$ the infinite series inside the parentheses diverges, so it will eventually pass $100,000$ .

So, even though it will take a ridiculously long time, $\boxed{\text{yes}}$ he will eventually make it! :)

For those that are still confused by the apparent paradox, remember that it's streched uniformly so some of the streching occurs behind the snail at any time too.

Anthony Holm - 4 years, 7 months ago

Ah yes... Good point!

Geoff Pilling - 4 years, 7 months ago

Yes , I totally agree 😃

Sabhrant Sachan - 4 years, 7 months ago

I think if the chord stretches continuously throughout the one minute it takes $e^{100000}-1$ minutes

Wen Z - 4 years, 7 months ago

I knew it was large, but how did you come up with that number? :^)

Geoff Pilling - 4 years, 7 months ago

here

Wen Z - 4 years, 7 months ago

Ah yes. I love this question! It also shows us how to deal with the generalized scenario, and how to modify the scaling so that the snail cannot reach the end :)

Calvin Lin Staff - 4 years, 6 months ago

Poor snail!

This is not an original problem, but was presented to me by my dad, Richard Pilling. :-) I understand it's a Martin Gardner original.

1 solution

0 pending reports