$A$ girl running a race accelerates at $2.5 \, m s^{-2}$ for the first $4 \, s$ of the race. How far does she travel in this time (in $, m$)?

$(a)$ Derive the second equation of motion $S = ut + \frac{1}{2}at^2$ graphically,where the symbols have their usual meanings.
$(b)$ $A$ car accelerates uniformly from $18 \text{ km h}^{-1}$ to $36 \text{ km h}^{-1}$ in $5 \text{ s}$. Calculate the acceleration and the distance covered by the car in that time.

Draw a velocity-time graph for a body which starts to move with initial velocity $u$ under a constant acceleration $a$ for time $t$. Using this graph,derive an expression for the distance covered $S$ in time $t$.

$A$ trolley,while going down an inclined plane,has an acceleration of $2 \,m \,s^{-2}$. What will be its velocity (in $cm \,s^{-1}$) $3 \,s$ after the start?

Starting from rest at the top of an inclined plane, a body reaches the bottom of the inclined plane in $4$ seconds. In what time does the body cover one-fourth of the distance starting from rest at the top?

$A$ cyclist driving at $36 \, km \, h^{-1}$ stops his cycle in $2 \, s$ by the application of brakes. Calculate $(i)$ retardation $(ii)$ distance covered during the application of brakes.