Beetle's meeting up

[BMAD]

A high quality rubber band is fastened and hung from a horizontal pole with a cannonball at its end. Two facing ladybugs are crawling along this rubber band toward each other. From their respective starting positions (8 cm apart), each small beetle crawls toward the other at a speed of 1 cm per second. However, in the length of time each beetle crawls 1 cm, the cannonball, thanks to the force of gravity, stretches the rubber band an additional 8 cm. Will the poor ladybugs ever meet? And, if yes, when? If not, why?!

[harey]

Yes, and the beetle heading down will even reach the cannonball.

As I have already seen the problem, I will refrain from posting.

[harey]

I think it is a pity that the solution is not posted when nobody finds it. So here we go:

Just integrate. As I safely integrate only x=k and x=e**x, I will leave the developement to someone else. (Anyway, there were "Best Answers" for less than that.)

The cannonball is 8 times faster than the bugs. Divide the band in more than 8 equal parts and mark the ends of these parts. Two adjacent marks will move away from each other by a speed that is smaller than that of the bugs. So the bugs will reach the first amrk, the second... and they will eventually meet (and if they contimue, one will reach the cannonball and the second one the fixation.

The speed of the cannonball is constant. The speed of the bugs is increasing.

Increasing speed is not enough. The bugs are accelerating and the acceleration is increasing.

Edited December 18, 2013 by harey

[bonanova]

goes down the band in fixed steps that become a decreasing fraction of its length. But the fraction decreases slowly, as 1/n. This is the harmonic series, and it diverges. That is, its partial sums increase without limit. That is, the top bug eventually reaches the ball and, before that, he reaches the other bug.