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 March 3rd, 2018, 02:45 AM #1 Newbie   Joined: Mar 2018 From: poland Posts: 2 Thanks: 0 How to solve this problem? (HARD) A giant rabbit is tied to a pole in the ground by an infinitely stretchy elastic cord attached to its tail. A hungry flea is on the pole watching the rabbit. The rabbit sees the flea, jumps into the air and lands one kilometre from the pole (with its tail still attached to the pole by the elastic cord). The flea gives chase and leaps into the air landing on the stretched elastic cord one centimetre from the pole. The rabbit, seeing this, again leaps into the air and lands another kilometre away from the pole (i.e., a total of two kilometres from the pole). Undaunted, the flea bravely leaps into the air again, landing on the elastic cord one centimetre further along. Once again the rabbit jumps another kilometre and the flea jumps another centimetre along the cord. If this continues indefinitely, will the flea ever catch up to the rabbit? (Assume the earth is flat and extends infinitely far in all directions.)
 March 3rd, 2018, 05:41 AM #2 Global Moderator   Joined: Dec 2006 Posts: 19,887 Thanks: 1835 Yes.
 March 3rd, 2018, 08:37 AM #3 Senior Member   Joined: Sep 2016 From: USA Posts: 504 Thanks: 283 Math Focus: Dynamical systems, analytic function theory, numerics In the limit they will be $-\frac{1}{12}$ meters apart.
 March 3rd, 2018, 08:54 AM #4 Newbie   Joined: Mar 2018 From: poland Posts: 2 Thanks: 0 Thank you, but If could I ask you, HOW did you solve it?
March 3rd, 2018, 09:31 AM   #5
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Quote:
 Originally Posted by SDK In the limit they will be $-\frac{1}{12}$ meters apart.
a negative length?

 March 3rd, 2018, 02:14 PM #6 Global Moderator   Joined: Dec 2006 Posts: 19,887 Thanks: 1835 It's an easy question, but first consider what happens if the rabbit leaps a much shorter distance, such as 2 cm. Bear in mind that once the flea is on the elastic, it moves as the elastic is stretched by the rabbit's leap. Thanks from topsquark
May 7th, 2018, 03:17 PM   #7
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Quote:
 Originally Posted by mansmath A giant rabbit is tied to a pole in the ground by an infinitely stretchy elastic cord attached to its tail. A hungry flea is on the pole watching the rabbit. The rabbit sees the flea, jumps into the air and lands one kilometre from the pole (with its tail still attached to the pole by the elastic cord). The flea gives chase and leaps into the air landing on the stretched elastic cord one centimetre from the pole.
So after one jump the rabbit is 999.98 meters away from the flea.

Quote:
 The rabbit, seeing this, again leaps into the air and lands another kilometre away from the pole (i.e., a total of two kilometres from the pole).
So the cord has doubled in length. The flea, that was 1 cm from the pole is now 2 cm from the pole.
Quote:
 Undaunted, the flea bravely leaps into the air again, landing on the elastic cord one centimetre further along.
The flea is now 3 cm from the pole and the rabbit is 1999.97 meters from the flea.

Quote:
 Once again the rabbit jumps another kilometre and the flea jumps another centimetre along the cord.
The cord has stretched by 3/2 so, before the jump the flea was 9/2= 4.5 cm from the pole and after jumping one cm. is 5.5 cm from the pole, 2994.5 m from the rabbit.
Quote:
 If this continues indefinitely, will the flea ever catch up to the rabbit? (Assume the earth is flat and extends infinitely far in all directions.)
The sequence is 999.8, 1999.97, 2994.5, etc. No, that is NOT converging to 0.

 May 8th, 2018, 05:16 AM #8 Global Moderator   Joined: Dec 2006 Posts: 19,887 Thanks: 1835 If you obtain a formula for those distances, you'll see that the flea does eventually catch up.
 May 8th, 2018, 08:34 PM #9 Senior Member   Joined: Sep 2016 From: USA Posts: 504 Thanks: 283 Math Focus: Dynamical systems, analytic function theory, numerics I think CountryBoy is not assuming that each jump made by the rabbit stretches the elastic cord and causes the flea to move further away from the pole. This happens in addition to the distance traveled by the flea. His analysis is correct within that interpretation. However, I think this is not the interpretation intended. Math is precise for a reason, though rarely is that reason so transparent as attempting to get inside the mind of a random internet crazy person to determine what they mean when they say a rabbit leaps 1km.
 May 8th, 2018, 09:24 PM #10 Global Moderator   Joined: Dec 2006 Posts: 19,887 Thanks: 1835 Country Boy stated "So the cord has doubled in length. The flea, that was 1 cm from the pole is now 2 cm from the pole." Hence the flea's movement due to the cord being stretched by the rabbit is being taken into account. This is an old puzzle.

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