# Thread: the ant on a rubber rope

1. ## the ant on a rubber rope

I cannot solve the problem of the ant on a rubber rope (Ant on a rubber rope - Wikipedia, the free encyclopedia)
in the case of a rope stretching exponentially:

the target end of the rope y(t) goes away exponentially

y(t)= D*exp(H*t)

where D and H are positive constants. D is the initial length of the rope to be crossed by the ant.

Given that the rope stretches uniformly, I believe that the position of the ant on the rope x(t) should evolve according to

dx(t)/dt=c+x(t)*exp(H*t)

where c is the constant proper speed of the ant irrespective of rubber band stretching
initial condition x(0)=0

Can the ant reach the end of the rope depending on the relative values of c, D and H?
is it possible to prove that the ant will never reach the target, whatever the values of the constants?
or is it a critical point on the rope beyond which the target cannot be reached

Denis

2. ## Re: the ant on a rubber rope

Consider the problem in coordinates which get stretched together with the rubber band:
dy(t)/dt = c/exp(H t) = c exp(-H t)
This is easy to solve, and the total distance of the ant approaches c/H. For c>H, it reaches the other side, for c<=H, it does not.

3. ## Re: the ant on a rubber rope

Originally Posted by mfb
Consider the problem in coordinates which get stretched together with the rubber band:
dy(t)/dt = c/exp(H t) = c exp(-H t)
This is easy to solve,
.

thank you for your answer; I don't see precisely the equation to be solved, would you be kind enough writing it

Originally Posted by mfb
and the total distance of the ant approaches c/H. For c>H, it reaches the other side, for c<=H, it does not.
for any value of c and H, does the ant win the race if D<c/H ?

4. ## Re: the ant on a rubber rope

dy(t)/dt = c exp(-H t)
This can be solved by integration of both sides from t=0 to t=T (or infinity).

I missed the additional D in your formula, but this is just a scaling of c:
dy(t)/dt = c/D exp(-H t)

The ant reaches the other side for c/(DH)>1. In other words, the initial velocity of the ant has to be larger than the initial extension velocity of the rubber band.

thanks!