- #1

doktorwho

- 181

- 6

## Homework Statement

The functions are given:

##r(t)=pe^{kt}##

##\theta (t)=kt##

##v(r)=\sqrt2kr##

##a(t)=2k^2r##

Find the radius of the curvature of the trajectory in the function of ##r##

## Homework Equations

$$R=\frac{(\dot x^2 + \dot y^2)^{3/2}}{(\dot x\ddot y - \ddot x\dot y)}$$

There is also a second equation:

$$R=\frac{(1- y'^2)^{3/2}}{y''}$$

## The Attempt at a Solution

I tried using the first one to get the dependence of ##t## and then transforming to the dependence of ##r## but i get stuck. Here:

##R=\frac{(\dot x^2 + \dot y^2)^{3/2}}{(\dot x\ddot y - \ddot x\dot y)}##

I did not know what exactly are ##x, y## in my problem statement so i supposed that they are the radial and angle component of the velocity vector.

##v(t)=pke^{kt}\vec e_r + pke^{kt}\vec e_{\theta}##

So the ##(\dot x^2 + \dot y^2)^{3/2}=(2p^2k^2e^{2kt})^{3/2}##

##=\sqrt2pke^{kt}##

I have continued like this and used the acceleration in the polar coordinates for the below part but fail to get anything. Is my thinking from the start wrong? Could i have used an easier way? Perhaps finding the dependence on ##r## immediately from the result insted of first from ##t##?