A spacecraft leaves a circular orbit of radius and follows the three–dimensional…

MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

A spacecraft leaves a circular orbit of radius

20\text{ km}

and follows the three–dimensional trajectory

\boldsymbol r(t)=\bigl\langle 20\cos(bt),\,20\sin(bt),\,k\sin^{2}(bt)\bigr\rangle,

where

b>0

k

are real constants and

t

is measured in seconds. (a) Compute the velocity vector

\boldsymbol v(t)

in terms of the parameters

b

and

k

. (b) Determine the value of

k

(in terms of

b

or otherwise) for which the spacecraft moves at constant speed for all

t

. (c) For the value of

k

found in part (b), find the exact constant speed of the spacecraft. (d) For that same value of

k

, decide whether the spacecraft’s velocity is ever parallel to its position vector. Justify your conclusion rigorously.