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I came across the following question in a textbook:

A horse is attached with a infinitely stretchable rope to a pole. At time 0 the rope is 2m long. On the rope walks a snail with a speed of 0.01m/s to the horse., at time 0 the snail is next to the pole. After every full minute the horse instantaneously jumps 100m away from the pole. This process continues until the snail reaches the horse.

a) How far is the snail from the rope after 1 minute, 2 min, 3min.

b) Will the process ever finish?

The first part is easy:

1 minute before jump: $0.01m$
2 minutes before jump: $\frac{102}{2}*0.01 + 0.01 = 0.52$
3 minutes before jump: $\frac{202}{102}*0.52 + 0.01 \approx 1.0398$

I can't find a function formula for the distance traveled in function of the time, finding one for the length of the rope is easy: $l(t) = 2 + 100t$. I plotted the values on my TI-84 with a program (the time is on the x-axis):
The distance travelled
The fraction of the rope travelled

The distance walked by the snail looks linear (but is isn't) and the relative speed of the snails seems to decrease, but I can't find the answer to (b).

I found this similar question, but in that problem the horse continuously moves away from the pole. Based on the context I also think it should be possible to solve the problem without integrals.

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KarelPeeters
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  • Try relating it to the original problem by setting the horses average velocity and stating the horse moves continuously. Then think about what error this method produces per minute and add it back as a correction factor. (a core reason behind why this process ends is that the length of rope both behind and ahead of the snail stretches so if the snail ever gets passed halfway then the distance traveled increases faster then the distance left to travel) – shai horowitz May 13 '16 at 21:25
  • @shaihorowitz Taking a closer look at the answer over in that question, I don't understand on of the fist formulas he descibes: "the fraction of the total band covered by the snake is at time t is $\frac{u}{l+vt}". Could you help me? I also still think that the problem should be solvable without integrals, based on where I found the problem. I also don't know integrls yet. And about you reason why it should end, does it really work that way? Wont the relative speed of the snail become so low after a while that it doesn't matter any more? – KarelPeeters May 14 '16 at 08:48

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