About the "ant on the rubber band paradox" I'm reading the solutions for the ant on the rubber band paradox and I'm not quite sure if I understand this step
this step
  correctly.
Can somebody please elaborate on the part that involves φ? What is it actually? Why are you able to change X to alpha(the speed of the ant)? I know that it's something about the relative position and speed but I still don't quite understand how you can replace X with alpha.
The solution I've read (via Wikipedia: https://en.wikipedia.org/wiki/Ant_on_a_rubber_rope)
 A: Imagine there are closely-spaced markings all along the rubber band, each labeled with a number.
At the start, each number shows the distance of its mark from the start of the rubber band, measured in kilometers.
So the mark at the far left end is labeled $0,$ the mark at the far right end is labeled $1,$ the mark in the exact center is labeled $0.5$, the mark one quarter of the way from the start is labeled $0.25$, the mark one centimeter to the right of the $0.25$ mark is labeled $0.25001$, and so forth.
As the rubber band stretches, it stretches uniformly so that each mark stays at the same fraction of the length of the rubber band.
That is, the $0$ mark is always at the far left, the $1$ mark is always at the far right,
the $0.5$ mark is always exactly halfway between the $0$ and $1$ marks, the $0.25$ mark is always exactly halfway between the $0$ and $0.5$ marks, and so forth.
So when the rubber band is two kilometers long, the $0.25001$ mark is two centimeters to the right of $0.25$, because every distance between marks got proportionally longer.
The marks on the rubber band represent the coordinate $\psi$ in
the Wikipedia solution of the continuous version of the problem
(where one end of the rubber band is being pulled continuously outward at a constant speed, and we also assume the ant moves continuously at a constant speed relative to whatever point on the rubber band it is on at any given moment).
To make this really work as a coordinate, it's better if we imagine that there are not just many markings along the rubber band; we should imagine that every point on the rubber band is marked and labeled with a number.
The Wikipedia solution then introduces another symbol, $\phi(t),$ defined as the $\psi$-coordinate of the ant at  time $t$;
that is, if at $t$ seconds after things start moving you look at the point on the rubber band where the ant is standing and read the
$\phi(t)$ is the label on the mark at that point on the rubber band.
Then $\phi'(t)$ is the rate at which "the number on the mark where the ant is" is increasing at $t$ seconds after the start.
At the instant everything starts moving, at $t=0$, the length of the rubber band is $c$ and the ant is moving at a speed $\alpha,$ so
$\phi'(0) = \frac\alpha c.$
After one second, at $t=1$, the length of the rubber band is $c+v,$
so the number on the marks the ant passes is increasing at a proportionally slower rate, $\phi'(1) = \frac\alpha{c+v}.$
More generally, after $t$ seconds for any $t > 0,$ the length of the rubber band is $c+vt$ and $\phi'(1) = \frac\alpha{c+vt}.$
