Mathematics

Problem

Consider the polar curve

r(\theta)=1+2\cos \theta

. A particle moves along this curve so that its angular position satisfies

\theta(t)=2t

for

0\le t\le2\pi

. The origin is fixed. (a) Express the velocity vector

\bigl(\dfrac{dx}{dt},\dfrac{dy}{dt}\bigr)

in terms of

t

. (b) Calculate the particle’s speed at

t=\dfrac{\pi}{3}

. (c) Determine all values of

t

in the interval

0\le t\le \pi

for which the velocity vector points directly toward or directly away from the origin. Justify your conclusion.