## Particle Moving Along y Axis on Surface of Rotating Ellipse

Suppose an ellipse with one focus at the origin is made to rotate about the origin in the\[xy\]

plane. A particle on the surface of the ellipse is made to move on the surface of the ellipse as the ellipse rotates. What will be the velocity and acceleration of the particle?If the minor and major axes have length

\[b, \; a\]

respectively, the coordinates of a point on the surface of the ellipse with origin at one focus is \[(r, \theta )\]

in polar coordinates with \[r= \frac{b^2}{a(1- ecos \theta )}\]

and the velocity is \[\frac{dr}{dt}= - \frac{eb^2 \dot{\theta} sin \theta }{a(1-e cos \theta )^2} \]

.The acceleration is

\[\frac{d^2r}{dt^2}= - \frac{eb^2 (\ddot{\theta} sin \theta + \dot{\theta}^2 cos \theta )}{a(1-e cos \theta )^2} - \frac{2e^2b^2 (\dot{\theta}^2 sin^2 \theta}{a(1-e cos \theta )^3} \]

.