A spacecraft travels in a plane using a guidance system that reports its velocity in…

MATHEMATICS · Multi-part · AP · AP Precalculus · Functions with Parameters, Vectors, and Matrices · Functions with Parameters, Vectors, and Matrices

Problem

A spacecraft travels in a plane using a guidance system that reports its velocity in polar coordinates relative to a fixed origin. For

0\le t\le1

hour, the radial and angular velocity components are

\frac{dr}{dt}=5t\quad\text{(kilometers per hour)},\qquad\frac{d\theta}{dt}=t^{2}\quad\text{(radians per hour)},

where

r(t)

is the distance from the origin in kilometers and

\theta(t)

is the polar angle measured from the positive

x

–axis. At

t=0

, the spacecraft is located at polar coordinates

(r,\theta)=(2,0)

. (a) Determine

r(t)

for

0\le t\le1

hour. (b) Determine

\theta(t)

for

0\le t\le1

hour. (c) Express the spacecraft’s position at

t=1

hour in Cartesian coordinates

\langle x,y\rangle

(kilometers). (d) Compute the spacecraft’s speed at

t=1

hour and interpret this value in the context of the problem.