Mathematics

Problem

An engineering student models a mechanical part using an ellipse defined by

\frac{x^{2}}{25} + \frac{y^{2}}{9} = 1

. A particle moves along the auxiliary circle such that its position is given by

Q(5\cos(t), 5\sin(t))

, where

t

is time in seconds. A second particle

P

moves along the ellipse such that

P

is always the vertical projection of

Q

onto the ellipse. Find the speed (magnitude of velocity) of particle

P

at time

t = \frac{\pi}{3}\,\text{seconds}