For a twice-differentiable position vector

MATHEMATICS · Single correct · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

For a twice-differentiable position vector r(t)=⟨x(t),y(t)⟩\mathbf r(t)=\langle x(t),y(t)\rangle, a student writes in their notes: "Speed =[x′′(t)]2+[y′′(t)]2=\sqrt{[x''(t)]^{2}+[y''(t)]^{2}}." Which of the following statements best evaluates the student’s claim? (a) The claim is correct because speed is the magnitude of acceleration. (b) The claim is incorrect; speed equals x′(t)+y′(t)x'(t)+y'(t), the sum of the component velocities. (c) The claim is incorrect; speed is the magnitude of velocity and should be [x′(t)]2+[y′(t)]2\sqrt{[x'(t)]^{2}+[y'(t)]^{2}}. (d) The claim is correct only when the particle moves along a straight line.

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