$($I$)$ A particle moving in one dimension has the trajectory, $x(t),$ shown in the adjacent figure: Time is measured in seconds and position in metres.

$($a$)$ List the time$($s$)$ at which the particle is at the origin, i$.$e$.,x(t_{**})=0m.$

$t=5,t=10+{10}^7$

$($b$)$ Describe the time$($s$)$ when the particle is moving forward $[$i$.$e$.,$ toward larger values of $x.$

$t=$I,$t=3(1,3)$ and $ac0$

$($c$)$ Describe the time$($s$)$ when the particle is at rest.

$t=(0,1),(6,9)$

$($d$)$ Compute the average velocity of the particle during the time intervals listed below.

$($i$)t=0s$ to $t=10s$

$($ii$)t=1s$ to $t=5s$

$\frac{0-2}{10-0}=\frac{-\frac{2}{10}\frac{?}{2}}{\frac{x(10)-x(0)}{10-0}}=$

$0-2$

$($iii$)t=3s$ to $t=6s$

$\frac{0-2}{s-1}=\frac{-2}{4}=-\mathrm{.}3\frac{m}{s}$

$($iv$)t=6s$ to $t=9s$

$\frac{-2-4}{6-3},\frac{-2}{3}\frac{m}{s}$

Math end $\frac{0\sqrt[\phantom{2}]{2}}{2}$

$\frac{-2-2}{4-6}=\frac{-4}{3}\frac{m}{s}$

Estimate the instantaneous velocity of the particle at the following times.