Problem $1$

A velocity field is given by$,$

vec$(V)(x,y,t)=\{\begin{array}{ccc}-1\text{h}\text{a}\text{t}& ()-1\text{h}\text{a}\text{t}()\text{f}\text{o}\text{r}& 0t1\\ -1\text{h}\text{a}\text{t}& ()+2\text{h}\text{a}\text{t}()\text{f}\text{o}\text{r}& 12\end{array}$

where $x,y$ are in $[$m$],t$ is in $[$s$],$ and vec$(V)$ is in $[$m$/$s$].$ In the sketches required for the following parts, include the region $-4x4$ and $-4y4$ in the drawings and identify important features.

$($a$)$ Sketch the streamline that passes through the point $(x,y)=(0,0)$ as it would appear at time $t=1\mathrm{.}5.$

$($b$)$ Sketch the pathline of a particle released from the point $(x,y)=(0,0)$ at time $t=0$ as it would appear at time $t=4.$

$($c$)$ Sketch the streakline of particles released from the point $(x,y)=(0,0)$ as it would appear at time $t=4.$