I'm doing these examinations but the lecturer has changed one or two of the questions since I originally sat the course so they are not in my notes how to approach. This is an example of one he changed from a tutorial sheet he gave out a few months back(adsbygoogle = window.adsbygoogle || []).push({});

"A particle acted upon by a central attractive force [tex]u^{3}\mu[/tex] is projected with a velocity [tex]\frac{\sqrt{\mu}}{a}[/tex] at an angle of [tex]\frac{\pi}{4}[/tex] with its initial distance [tex]a[/tex] from the centre of the force; show that its orbit is givin by [tex]r=ae^{-\theta}[/tex] "

I'd have already derived the formula [tex]\frac{d^{2}u}{d\theta^{2}}+u=\frac{F(\frac{1}{u})}{h^{2}u^{2}}[/tex] where [tex]r^{2}\dot{\theta}=h[/tex] and [tex]u=\frac{1}{r}[/tex]

But I'm used to doing these types of problems where the particle is projected from an apse. Where I get stuck is when I've gotten it down to the differential equation solution I have:

[tex]u=Ae^{\theta}+Be^{-\theta}[/tex]. When [tex]\theta=\frac{\pi}{4}[/tex]; [tex]u=\frac{1}{a}[/tex]. So I've got [tex]Ae^{\frac{\pi}{4}}+Be^{-\frac{\pi}{4}}=\frac{1}{a}[/tex] But now what? How do I solve for [tex]A[/tex] and [tex]B[/tex]? Usually I'd be able to say that [tex]\frac{du}{d\theta}=0[/tex] but since this isn't at an apse I can't do that (at least I dont think I can). So how can I get my second simultaneous equation to solve the arbitrary constants A and B?

Thanks

-Declan

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# Homework Help: Mechanics, central orbit/force question

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