A particle moves in the plane so that

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

Problem

A particle moves in the plane so that, at time

t\ge 0

, its position is described by polar coordinates

(r,\theta)

, where

r=f(\theta(t))

for a differentiable function

f

and a differentiable angular position

\theta(t)

. The origin is fixed and the standard polar-to-Cartesian relationships apply. (a) Write

x

and

y

explicitly as functions of

r

and

\theta

. Then rewrite each solely in terms of

f\bigl(\theta(t)\bigr)

and

\theta(t)

. (b) Derive formulas for

\dfrac{dx}{dt}

and

\dfrac{dy}{dt}

that involve

\dfrac{dr}{dt}

\dfrac{d\theta}{dt}

r

, and

\theta

. Clearly indicate every differentiation step. (c) Interpret the two distinct contributions you obtained in part (b) to the particle’s velocity. Describe how each contribution affects the particle’s motion relative to the origin.

Related theory

Polar Functions
Polar Coordinates
Trigonometric Functions

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