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ad absurdum

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## Homework Statement

A particle P of mass [tex]m[/tex] moves under the infulence of a central force of magnitude [tex]mkr^{-3}[/tex] directed towards a fixed point O. Initially [tex]r=a[/tex] and P has a velocity [tex]V[/tex] perpendicular to OP, where [tex]V^2 < \frac{k}{a^2}[/tex]. Prove that P spirals in towards O and reaches O in a time

[tex]T = \frac{a^2}{\sqrt{k-a^2V^2}}[/tex].

## Homework Equations

[tex]\frac{d^2u}{d\theta^2} - (\frac{k}{a^2V^2} - 1})u = 0[/tex]

## The Attempt at a Solution

I've got the equation [tex]r = a sech (\sqrt{\frac{k}{a^2V^2} - 1}) \theta}[/tex], which I think is right, but I have no idea how to find [itex]T[/itex] 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.