A particle moves along a plane curve for

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

Problem

A particle moves along a plane curve for 0≤t≤π0\le t\le \pi. Its acceleration vector is a(t)=⟨sin⁡t,  cos⁡t⟩\displaystyle \mathbf a(t)=\langle \sin t,\; \cos t\rangle. The following initial information is known: • Velocity at t=πt=\pi is v(π)=⟨0,2⟩\mathbf v(\pi)=\langle 0,2\rangle. • Position at t=0t=0 is r(0)=⟨3,−1⟩\mathbf r(0)=\langle 3,-1\rangle. Respond to all parts, showing complete justification. (a) Find the velocity vector v(t)\mathbf v(t) for 0≤t≤π0\le t\le \pi. (b) Determine the position vector r(t)\mathbf r(t) for 0≤t≤π0\le t\le \pi. (c) Compute the average velocity of the particle on the interval 0≤t≤π0\le t\le \pi. Express your answer as a vector. (d) Find the magnitude of the displacement of the particle from t=0t=0 to t=πt=\pi. Include units in your interpretation.

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