A particle moves along the curve described for by x(t)= t^ 3 3

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

Problem

A particle moves along the curve described for

t\ge 0

x(t)=\frac{t^{3}}{3},\qquad y(t)=\frac{t^{2}}{2}.

The particle is to be reparameterized so that it travels at the constant speed of

5

units per unit time while still tracing the same set of points in the same direction. (1) Write an equation that relates the new parameter

s

(with

s=0

corresponding to

t=0

) to the original parameter

t

by equating arc length along the curve with

5s

. Clearly specify all limits of integration. (2) Solve the equation from part (a) to express

t

as a function of

s

. Simplify your result as much as possible. (3) Using your result from part (b), determine the value of the new parameter

s

at which the particle has reached the point corresponding to the original parameter value

t=4

. Provide your answer to three decimal places. (4) How long (in units of the new parameter

s

) will it take the particle to travel a total distance of

20

units along the curve?