ad absurdum
Feb26-08, 03:52 PM
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.
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.