Mathematics

Problem

A particle moves along the

x

–axis with velocity

v(t)=kt^{2}-6t+4

, where

k

is a real constant and

t

is measured in seconds for

t\ge 0

. The particle’s position function

s(t)

(in meters) satisfies

s(0)=2

. After 3 seconds the particle is observed to be 10 meters from the origin, so that

s(3)=10

. (a) Derive an explicit expression for

s(t)

in terms of

k

. (b) Determine the value of the constant

k

. (c) Using your value of

k

, write the fully determined position function

s(t)

. (d) Find all times in the interval

0\le t\le 3

when the particle changes direction, and justify your answer.