A question about a route of a point that travels in a particular way through the plane I don't know exactly how to classify this question. It is not from any homeworks, just something I've been wondering about.
Let's say that in the beginning of an experiment ( the beginning is $t=0$ secs) we have two points on the plane: one on $(0,0)$ and one on $(0, 1)$.
They start moving by the following rules:


*

*The point that was on $(0,0)$ moves right, on the $x$-axis, in a constant speed of $1~\text{m}/\text{s}$.

*The point that was on $(0,1)$ also moves in a constant speed of $1~\text{m}/\text{s}$, but its direction changes, so that the direction of its speed is always directed to the first point (like a missile that follows a moving object).
I hope I made myself clear. 
My question is: Is there a nice formula for the location of the second point on the plane for a given time $t$? How can it be derived? And what if we change the ratio between the constant speeds?
Thank you for your time reading my question...

Let the first point's position be $(t,0)$, and the second $(x(t),y(t))$.
We have that the direction of the second point is proportional to the difference between the points:
$$(x'(t), y'(t))\propto(t - x(t), -y(t))$$ 
But, since the speed is constant, we must have that:
$$(x'(t), y'(t))=\left(\frac{t-x(t)}{\sqrt{(t - x(t))^2 + y^2(t)}}, -\frac{y(t)}{\sqrt{(t - x(t))^2 + y^2(t)}}\right)$$ 
With $x(0) = 0, y(0) = 1$.
Now we must solve the differential equation.
Addendum
This equation does have a closed solution involving the Lambert $W$-function, which can be obtained by moving to polar coordinates, but it gets very ugly - If anyone can offer a simpler solution, I'd be glad to hear of one. The solution is also closely related to a similar problem, that of the Tractrix, which has constant distance rather than constant speed.
In any case, here is what the resulting curve looks like, for $t\in[0,1]$:

