# Cotes' Spiral

1. Feb 26, 2008

1. The problem statement, all variables and given/known data
A particle P of mass $$m$$ moves under the infulence of a central force of magnitude $$mkr^{-3}$$ directed towards a fixed point O. Initially $$r=a$$ and P has a velocity $$V$$ perpendicular to OP, where $$V^2 < \frac{k}{a^2}$$. Prove that P spirals in towards O and reaches O in a time

$$T = \frac{a^2}{\sqrt{k-a^2V^2}}$$.

2. Relevant equations
$$\frac{d^2u}{d\theta^2} - (\frac{k}{a^2V^2} - 1})u = 0$$

3. The attempt at a solution
I've got the equation $$r = a sech (\sqrt{\frac{k}{a^2V^2} - 1}) \theta}$$, which I think is right, but I have no idea how to find $T$ from this. I haven't covered hyperbolic functions in much detail before (which is a shame, because they are assumed on this course) so I may be missing something obvious. I'm guessing I should be evaluating some integral but I can't think of anything/see anything useful in my notes. Any hints would be much appreciated.