The graph given below is the distance$-$time graph of an object.

$(i)$ Find the speed of the object during first four seconds of its journey.

$(ii)$ How long was it stationary ?

$(iii)$ Does it represent a real life situation ? Justify your answer.

A circular cycle track has a circumference of $314\, m$ with $A B$ as one of its diameter. $A$ cyclist travels from $A$ to $B$ along the circular path with a velocity of constant magnitude $15.7\, m s ^{-1}$. Find the

$(a)$ distance moved by the cyclist.

$(b)$ displacement of the cyclist, if $A B$ represents north$-$south direction.

$(c)$ the average velocity of the cyclist.

Derive following equations for a uniformly accelerated motion

$(i)$ $v=u+a t$

$(ii)$ $S=u t+1 / 2 a t^{2}$

$(iii)$ $v^{2}-u^{2}=2 a S,$ where symbols have their usual meaning.

An object starts a linear motion with velocity $'u^{\prime}$ and with uniform acceleration ' $a^{\prime}$, it acquires a velocity $'v^{\prime}$ in timet

$(a)$ Draw its velocity$-$time graph.

$(b)$ Obtain Ist equation of motion, $v=u+a t,$ for velocity $-$ time relation by using velocity$-$time graph.

$(c)$ A body moving with a velocity of $2\, m s ^{-1}$ acquires a velocity of $10 \,m s ^{-1}$ in $5\, s$. Find its acceleration.

A person standing at $A$ goes to $B$ by following any of the paths $1,2$ or $3 .$ Which path we can measure to find the average velocity ?