$($a$)$ For a body moving in a straight line with constant acceleration $a,$ initial displacement $s(0)=s_0,$ and initial velocity $v(0)=v_0,$ derive the following expressions for the dependency of the velocity and displacement with time $t$ :

$($i$)v(t)=v_0+at$

$($ii$)s(t)=s_0+v_{0}t+\frac{1}{2}a{t}^2$

$($iii$)$ Sketch the graphs of the functions $v(t)$ and $s(t)$ for the case

$($b$)$ A car, moving with constant acceleration, passes successively two road segments, each $12$ m long. Find the acceleration of the car a and the speed $v_0$ at the beginning of the first segment if the car passed the first road segment in time $t_1=1s$ and the second in $t_2=2s.$

$($a$)$ Derive an expression for the displacement vector $r(t)$ of a projectile launched from the origin at an angle $$ to the horizontal with an initial speed $v_0.$ Take the horizontal and vertical unit vectors to be i and $k$ respectively. Derive expressions for the following:

$($i$)$ Velocity vector $v(t).$

$($ii$)$ Displacement vector $r(t)$ of the projectile.

$($iii$)$ The trajectory $z(x)$ of the projectile in terms of the horizontal $(x)$ and vertical $(z)$ components of the displacement.

$($iv$)$ The maximal horizontal range $x_{\text{m}\text{a}\text{x}}$ of the trajectory before the projectile hits the ground $(0k)$ again.

$($b$)$ A signal flare is fired from a flare gun with initial speed $v_0$ at an angle of $45$ to the horizon. The flare starts to burn from the launch, burns for $5$ s and stops burning at the highest point of its trajectory. What is the value of the initial speed $v_0?$ Ignore the air resistance to the flare's motion.

$($a$)$ Briefly explain the concept of the work in the context of mechanics. Give the expression for the work in the case of a constant force acting on a particle. What units are used to measure work?

$($b$)$ The position $r$ of a particle at time $t$ is given by the rule $r(t)=0\mathrm{.}5sin(2t)i+2j.$ Find the value of the velocity $v$ at the initial time $t=0.$

$($c$)$ A homogeneous stick AB $,$ the ends of which can slide without friction along the horizontal plane OA and the vertical wall OB $,$ is held in equilibrium by the tension of the string OC $($see figure$).$ The stick is inclined to the horizontal plane at an angle $$ and the string at an angle $.$

$($i$)$ Find the tension of the string T when the overall mass of the stick is $m.$ Additional help: The length $l_1=|BC|$ and $l_2=|AC|$ can be expressed using the length of the string $l_s$ as follows

$l_1=l_s\frac{cos}{cos},l_2=l_s\frac{sin}{sin}$

$($ii$)$ For which relations between $$ and $$ is $T>0?$