A particle moves along a parametric curve with position functions and
MATHEMATICS · Single correct · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions
Problem
A particle moves along a parametric curve with position functions x(t) and y(t). Its speed is constant and non-zero for all t≥0. Which statement best explains why solving the equation s(t)=21s(T) yields the time when the particle has traveled half the total distance covered by time T?
(a) Because the particle moves in a straight line, the displacement equals the arc length, so setting the x-coordinate equal to half its final value finds the time.
(b) Because the particle’s speed is constant, but the distance is obtained by integrating the speed once more, s(t)=21vt2; therefore halving the distance corresponds to t=T/2.
(c) Because the integrand in the arc-length integral is constant, the accumulated length is proportional to t, so equating half the total length directly gives the corresponding time.
(d) Because the parameter t itself always measures distance traveled, the condition s(t)=21s(T) is automatically satisfied at t=T/2 regardless of speed.