Ant on a rubber string, mathematical series

In summary, the ant will get to the end of the string when the sum of the lengths of the segments connecting the two ends is equal to the length of the string.
  • #1
prehisto
115
0

Homework Statement


Ideal rubber stirng with length L=1km.Ant is takng a walk on the string with speed v=1cm/s
After every minute(Δt=60s) ,string is getting longer by ΔL=1km.
1)Will ant get to the end of string?
2)If yes,then how long it will take ?


Homework Equations


1) So i used series expansion
[itex]\frac{s}{l}[/itex]=[itex]\frac{vΔt}{L}[/itex]+[itex]\frac{vΔt}{L+ΔL}[/itex]+..=
vΔt∑[itex]\frac{1}{L+iΔL}[/itex]

The ant will get to the end when s/l=1
So i got that.

2)But I have problem figuring out how to get the time needed to get there.
I have tip to use integral test,but i do not understand how to use it,beacuse as far as i know,tests are used to test if series converges.
Please,help here ?



The Attempt at a Solution



 
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  • #2
Your series expansion can be simplified to
$$\frac{s}{l}=\frac{vΔt}{L}\Bigl ( {1\over 1} + {1\over 2} + {1\over 3}+ {1\over 4} ... \Bigr )$$
If you (mentally) replace the summation by an integration you get a hunch that this sum is very close to a logarithm.
The expression in brackets is a harmonic number link
Mathematicians will claim the ant gets there.
Physicists are much more realistic: they take the lifetime of the universe into consideration and claim it doesn't get there.
 
  • #3
BvU said:
Your series expansion can be simplified to
$$\frac{s}{l}=\frac{vΔt}{L}\Bigl ( {1\over 1} + {1\over 2} + {1\over 3}+ {1\over 4} ... \Bigr )$$
If you (mentally) replace the summation by an integration you get a hunch that this sum is very close to a logarithm.
The expression in brackets is a harmonic number link
Mathematicians will claim the ant gets there.
Physicists are much more realistic: they take the lifetime of the universe into consideration and claim it doesn't get there.

Are you sure about that? That series does not converge, I don't think.
 
  • #4
There is no question of convergence: the summation doesn't have to be continued ad infinitum. In fact, how far it has to be continued is OP part 2) of the exercise.
 
  • #5
I would like to offer a different analysis of this problem.

Let L represent the total length of the string at time t, and let r represent the rate at which the length is increasing (1 km/min). Then,
[tex]L=L_0+rt[/tex]
where L0 is the initial length (1 km). The velocity of the far end of the string is dL/dt = r. The velocity at location y along the string is
[tex]v_s(y)=r\frac{y}{L}=r\frac{y}{L_0+rt}[/tex]
When the ant is at location y, its velocity is
[tex]\frac{dy}{dt}=r_A+v_s(y)=r_A+\frac{ry}{L_0+rt}[/tex]
where rA is the velocity of the ant relative to the string.

The solution to this differential equation subject to the initial conditions is:
[tex]\frac{y}{L_0+rt}=\frac{r_A}{r}\ln \left(\frac{L_0+rt}{L_0}\right)[/tex]
The ant reaches the end of the string when [itex]y=L_0+rt[/itex]. Therefore, the time required is obtained from:

[tex]t=\frac{L_0}{r}\left(\exp(\frac{r}{r_A})-1\right)[/tex]

Chet
 
  • #6
Interesting factor ##\gamma## between the "("faster!") discrete case and the continuous case !

With ##\gamma = 0.57721566490153286060651209008240243104215933593992## the discrete ant gets there 3*10723 minutes before the continuous ant !

But I'm afraid we've lost our prehisto friend here. Still listening in ?
 
  • #7
I red the first post and that was enough for me at the time. And forgot about this thread :(
But thanks guys.
Now I am looking into latest posts and they are interesting!
 

1. What is an ant on a rubber string mathematical series?

The ant on a rubber string mathematical series is a classic mathematical problem that involves an ant walking along a rubber string that is stretched infinitely in both directions. The ant starts at the middle of the string and takes steps in a specific pattern, either doubling or halving the length of the previous step, depending on the direction it is heading.

2. How is the ant on a rubber string problem solved?

The ant on a rubber string problem is typically solved using mathematical formulas and equations. The key to solving this problem is understanding geometric and arithmetic series and using them to calculate the total distance the ant travels.

3. What makes the ant on a rubber string problem interesting?

The ant on a rubber string problem is interesting because it combines elements of geometry, arithmetic, and infinite series. It also requires creative problem-solving skills and a strong understanding of mathematical concepts.

4. Is there a real-life application for the ant on a rubber string problem?

While the ant on a rubber string problem may seem like a purely theoretical math problem, it actually has real-life applications in fields such as computer science and physics. It can be used to model and solve problems related to distance, time, and growth.

5. Can the ant on a rubber string problem be solved using computer programs?

Yes, the ant on a rubber string problem can be solved using computer programs and algorithms. In fact, many online calculators and software programs have been developed specifically for solving this problem and similar mathematical series problems.

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