Snail race

Question

One end of a chewing gum is stuck to a wall and the other is given to a boy. The boy grips his end and starts to walk away from the wall, stretching the gum into a thread. A moment after the the boy started to walk a snail was placed on the thread tightly near the wall and started to crawl along the thread away from the wall.

The little boy goes 222 times faster than the snail crawls.
The gum is ideal: it'll be stretching and thinning as long as the boy pulls it. It stretches uniformly along its length and will never rip.

In these circumstances, will the snail ever reach the other end of the gum?

Merc · Accepted Answer

The answer depends on force of resistence between snail and thread. If it's enough only for snail going by the thread then snail never reachs other end. But if resistence is so that snail could keep its position on the chewing gum then stretching of the gum couldn't prevent the snail to achive the end because snail's position on the chewing gum will be rise in proportion to speed of the boy and so it's unimpotant how long will be thread.

Where are you studing, Ant?

Orior · Answer

Wrong answer #1) No. The velocity of the boy is greater than the velocity of the snail, and they can never meet.

Wrong answer #2) Yes, the boy walks the entire circumference of planet Earth, and catches up on the snail.

Right answer) Yes, and my guess is 222 days, because its all relative.

Ant_222 · Answer

To Merc:

Force of risistance...
Didn't fully get what you meant.

«If it's enough only for snail going by the thread then snail never reachs other end»

How can it be enough to crawl, but not enough to «keep its position on the chewing gum»?

The snail crawls due to this force applied from the thread.

It is high enough for the snail to be able to crawl along the thread.
That is, if in the beginning we put the snail on the center of the gum, and the snail is sleepeng and doesn't crawl anywere, then, when the boy begins to stretch the gum, the snail will always sit on the thread's center, however long it is.

In other words, a motionless snail will always keep its relative position on the thread: ratio of lengths into which the snail divides the thread will be constant.

As to the initial conditions, yes, the snail will crawl. But at a very low speed relative to the point of thread it is on, the point itself also moving. Thus, in the beginning, the distance between the snail and boy will constantly increase, because the snail itself and the thread's point near the wall will move very slowly.

So, why are you sure the snail will reach the end?

«...because snail's position on the chewing gum will be rise in proportion to speed of the boy and so it's unimpotant how long will be thread.»

The boy's velocity is constant. The absolute snail's velocity is the sum of its own velocity and the boy's constant velocity multiplied by the part the thread the snail has crawled over. So, the snail will start to get closer to the boy only after it crawls ((Vb-Vs)/Vb) of the thread, where Vb and Vs are the velocities of the boy and the snail.

But will it ever crawl that long?

I study at MEPHI, but the problem has nothing to deal with this fact...

To Orior:
«Right answer) Yes, and my guess is 222 days, because its all relative.»

That was catch #222. And you got caught.

Calculating the exact time will require differential equations, which is not really fun.

Merc · Answer

Hmm... MIFI. Indeed, doesn't metter at all. I haven't deel with integrals, logoriphms, differencial equations, spherical horces in vacuum and so on for a long time. So thank you for limbering-up. I should think more seriously from the beginning.

Orior · Answer

Can do differential equations. I lost my deriving licence <img class='emoticon' style='Width:19px;Height:19px;' data-delaysrc='https://uat-englishforward-forums-backend.s3.us-west-2.amazonaws.com/Emoticons/1.gif' src='https://uat-englishforward-forums-backend.s3.us-west-2.amazonaws.com/Emoticons/1.gif' alt='Emotion: smile'/>

Ant_222 · Answer

«Can do differential equations. I lost my deriving licence»
Deriving licence, the licence that allows you to derive (differentiate)? Hmmm, that's a problem. Deriving without a licence is illegal. I wouldn't wish that was typo and you meant "driving"...

«I haven't deel with integrals, logoriphms,
differencial equations, spherical horces in vacuum and so on for a long time.»

That is NOT interesting to solve it by dint of differential equations because any poor student can do that. The problem is how to manage without them...

Vanyatka · Answer

Sure the snail will catch the boy. Need to have a look at it from the distance

Let's say the snail will not move at all. The distance ahead of snail and behind the snail will grow proportionally. It can be like 9/10 of lengh ahead and 1/10 behindall the time. However slow the snail moves it will change this proportion, sooner or later approaching the boy. In other words, with every step of the snail the boy have to give part of his speed to stretch the gum behind the snail. The distance between b and s will grow in the beginning, then stay the same for a moment and finally start to decline.

Merc · Answer

No, it’s wrong. And I understood it.

Let’s count it without philosophy and other ???????????????????.

Look at the figure below.

x – is position of the boy in time t

d – is position of the snail in time t

?x – distance which boy went for time ?t

?d – distance which snail went for time ?t

Vb = ?x/?t – velocity of the boy

Vs = ?d/?t – velocity of the snail

x/d = ?x/?d = n = 222

l = x – d - way which snail will have to go until it reach the boy. If this magnitude will be less then zero, then snail reach the boy, else it will not.

So, let’s check.

l = x – d = x – x/n = x (n-1)/n. That’s enough. We know that x > 0 and it will be only rise, so forget about reachness of the snail.

Orior · Answer

Always best to solve this in small steps.

Firstly, if the snail never moves, will it always be half way between the wall and the boy?

Merc · Answer

Orior
Always best to solve this in small steps.

Firstly, if the snail never moves, will it always be half way between the wall and the boy?

No. But it doesn't mean that snail could reach the boy.

Snail race

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