Mathematics

Problem

A particle moves along a straight line so that its position (in meters) at time

t

seconds is given by

s(t)=3t^3-18t^2+24t-5

for

0\le t\le 6

. (a) Calculate

s(0)

and

s(6)

, including units. (b) Determine the particle’s average velocity, in meters per second, over the interval

0\le t\le 6

. Show all work that leads to your result. (c) Compute the instantaneous velocity

v(t)

of the particle, and find all times

t

in the open interval

(0,6)

at which

v(t)

equals the average velocity found in part (b). Justify your answer(s). (d) Describe the physical significance of the value(s) of

t

obtained in part (c) in the context of the particle’s motion.

Intuitive explanation

This problem is about understanding motion through calculus: velocity tells you how fast position changes, and acceleration tells you how velocity changes. To find when the particle stops, reverses direction, or moves fastest, you'll take derivatives of the position function—the first derivative $s'(t)$ gives velocity, and the second gives acceleration. The key insight is that the particle changes direction exactly when velocity equals zero, so finding those critical times reveals the structure of its journey.

Intuitive explanation

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