A comet fragment follows the planar trajectory for measured in years

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

Problem

A comet fragment follows the planar trajectory

\mathbf r(t)=\langle e^{t}\cos t,\,e^{t}\sin t\rangle

for

t

measured in years, where distances are expressed in astronomical units (AU). (a) Compute

\mathbf v(t)

and

\mathbf a(t)

in simplest form. (b) Prove that the velocity vector

\mathbf v(t)

is never orthogonal to the position vector

\mathbf r(t)

for

t\ge 0

. (c) Express

\mathbf a(t)

as a linear combination

A(t)\,\mathbf r(t)+B(t)\,\mathbf v(t)

and find the scalar functions

A(t)

and

B(t)

. (d) Using your result from part (c), determine the tangential and normal components of the acceleration at

t=0

and interpret their physical meanings.