Solution

$($a$)$

The velocity function is the derivative of the position function.

s$=$f$($t$)=$ t$312$t$2+36$tv$($t$)=$

dsdt

$=3$t$224$t $+36$

$($b$)$

The velocity after $2$ s means the instantaneous velocity when t $=2,$ that is

v$(2)=$

dsdt

$$t $=2$

$=3(2)224(2)+36=$ m$/$s$.$

The velocity after $4$ s is

v$(4)=3(4)224(4)+36=$ m$/$s$.$

$($c$)$

The particle is at rest when

v$($t$)=,$

that is$,$

$3$t$224$t $+36=3($t$28$t $+12)=3($t $2)($t $6)=,$

and this is true when t $=($smaller t$-$value$)$ or t $=($larger t$-$value$).$ Thus, the particle is at rest after s $($smaller t$-$value$)$ and after s $($larger t$-$value$).$

$($d$)$

The particle moves in the positive direction when v$($t$)?><=0,$ that is$,$

$3$t$224$t $+36=3($t $2)($t $6)?><=0.$

This inequality is true when both factors are positive

$($t $>)$

or when both factors are negative

$($t $<).$

Thus the particle moves in the positive direction in the time intervals

t $<$ t$1=$

and

t $>$ t$2=.$

It moves backward $($in the negative direction$)$ when

t$1=<$ t $<=$ t$2.$