1. ## baby problem.

i'm not sure if this is a well known problem, or not, but anyway, the problem is:

we have a baby who starts in the center of a circle, and the mother walks around the edge of the circle. the baby always crawls directly towards the mother. the baby and mother have respective speeds $u$ and $v$ with $v < u$. The circle has radius $R$. How long does it take for the baby to reach its mother?

Yes, this is a variation of a classic problem.
And there is a back-door approach to its solution.

A baby starts in the center of a circle.
Its mother walks around the edge of the circle.
The baby always crawls directly towards the mother at a speed $u$,
and the mother always walks at a speed $v$ with $v < u$.
The circle has radius $R$.
How long does it take for the baby to reach its mother?

SInce the baby is always moving directly toward its mother,
. . the mother is neither moving away from nor moving closer to the baby.

It is as if the mother is standing still and the baby is crawling directly to her.

To crawl a distance $R$ at a rate of $u$, it will take $\frac{R}{u}$ time units.

3. sorry, that's not what i mean. i will rephrase the problem, i specifically mean that if the position of baby is given by $b(t)$ then we have that

$|\dot{b}(t)| = u$

More specifically i'm asking for a solution to the differential equation:

$\vec{\dot{b}}(t) = u\frac{(\vec{m}-\vec{b)}}{|\vec{m}-\vec{b}|}$

where $\vec{m} = (R\cos{\frac{v}{R}t}, R\sin{\frac{v}{R}t})$

4. sadly i've still not been able to solve this yet

having done a little research on "pursuit curves" it seems that this is a well known and well studied problem, and that the solution to it is "very difficult". supposedly A.S. Hathaway and F.V. Morley solved it in 1921.

it suffices to consider a circle with unit radius and just the ratio between the two speeds.

it's easy enough to solve with a numerical solution, but seems difficult to get results more than say 10 significant figures using runge-kutta methods and the like.

it would be nice if i could find a more efficient way of approximating the result (ie finding 40 sig figs or so), and then maybe be able to find a taylor series-type thing for the time for different ratios... so far i seem to have that for a ratio $a$, unit radius and mother speed of 1, the time taken is about:

$\frac{1}{a} + \frac{1}{2a^3} + \frac{1}{2a^5} + O(a^{-6})$

5. On the topic of pursuit curves, I've found the following book by Paul Nahin and enjoyable read.