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

\[xy\]

plane. A particle on the surface of the ellipse is made to move as the ellipse rotates. What will be the velocity and acceleration of the particle?(P ALIGN="CENTER">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

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

.