Motion along a stretching band

[desquee]

This was given as a problem in a calculus textbook I'm working through (apologies if this should have gone in the physics forum)

1. Homework Statement
An ant crawls at 1foot/second along a rubber band whose
original length is 2 feet. The band is being stretched at 1
foot/second by pulling the other end. At what time T, if ever,
does the ant reach the other end?
One approach: The band's length at time t is t + 2. Let y(t)
be the fraction of that length which the ant has covered, and
explain
(a) y' = 1/(t + 2) (b)y =ln(t + 2) -ln 2 (c) T = 2e -2.

Homework Equations

∫1/x dx = ln(x)

The Attempt at a Solution

Given a, I can get to b by integrating and finding the constant, and then to c by solving for y=1, but I'm stumped on how to get to explain a. y' seems to be the ant's speed over the length of the band, by I don't understand why that is so.

 

[Nathanael]

Calculus by Gilbert Strang? I read that one too.

y' is the rate of change of the fraction which has been traversed. Suppose the ant was not crawling: when the rubber band is being stretched, the fraction of the rubber band behind the ant would not change (because it also stretches).

 

[desquee]

Yup.

I see that y' is the rate of change of the fraction, but I'm still not sure why why it equals what it does.

 

[Nathanael]

Consider two special cases to try to get an intuition:

The case where the ant is not crawling (but starts out at some initial fraction y₀).
The case where the rubber band is a fixed length.

What would the equation for y' be in each of these (separate) cases?

 

[desquee]

Got it, Thanks.