- #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##?