Mathematics

Problem

A particle moves along the

y

-axis with acceleration

a(t) = 4\cos t\quad (\text{meters per second squared}),\quad 0 \le t \le \frac{3\pi}{2}

. At time

t = 0

the velocity of the particle is

v(0) = -1\,\text{m/s}

and its position is

y(0) = 0

. (a) Determine an explicit expression for the velocity

v(t)

of the particle. (b) Find all times in

0 \le t \le \frac{3\pi}{2}

at which the particle is at rest. (c) For each sub-interval determined by the times found in part (b), state and justify the direction of motion of the particle. (d) Calculate the total distance traveled by the particle on the interval

0 \le t \le \frac{3\pi}{2}

. Provide an exact value and a numerical approximation correct to three decimal places.

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