- #1
cr7einstein
- 87
- 2
Homework Statement
This is a physics olympiad problem; and I am still struggling with it. I will quote it here:
" A particle moves along a horizontal track following the trajectory $$r=r_{0}e^{-k\theta}$$, where $$\theta$$ is the angle made by the position vector with the horizontal. Recall that the velocity in polar coordinates is $$\frac{d\vec r}{dt}=\dot r \hat r+r \dot \theta \hat \theta$$. If at $$t=0, \theta=0$$; the velocity is $$v_{0}$$, Find $$\theta$$ dependence of velocity and the angle the velocity vector makes with the radial vector."
The answer-$$v(\theta)=v_{0}$$ and independent of $$\theta$$; and $$\alpha(\theta)=tan^{-1}(+_{-}\frac{1}{k})$$ and independent of $$\theta$$.
It makes sense, as the equation resembles a logarithmic spiral, and the results hold for it; but how do I prove it?
Homework Equations
The Attempt at a Solution
I tried differentiating it all I could, but I am always getting an extra $e^{-k\theta}$ in the term for velocity.
$$\vec v=\frac{d}{dt}(r_{0}e^{-k\theta}) \hat r + r_{0}e^{-k\theta}\frac{d\theta}{dt}=-kr_{0}e^{-k\theta}\dot \theta\hat r + r_{0}e^{-k\theta}\dot \theta \hat \theta=r_{0}e^{-k\theta}\dot \theta(-k\hat r+\hat \theta)--------(1)$$
At $$\theta=0$$,
$$\vec v= \vec v_{0}=r_{0}\dot \theta(-k\hat r+\hat \theta)------------------(2)$$But then how do I prove that (1) and (2) are the same; i.e. how do I get rid of $$e^{-k\theta}$$ in the first equation? Or do I have to approach it differently( but I want to stick to polar coordinates).
Please help; I am posting this after struggling for 2 straight days; and now I am completely frustrated. What's even more even annoying that it is only a 4 mark (2 marks for each part) problem.
Thanks in advance!