$A$ train is travelling at a speed of $90\, km \,h^{-1}$. Brakes are applied so as to produce a uniform acceleration of $-0.5\, m s^{-2}$. Find how far (in $m$) the train will go before it is brought to rest.

$(a)$ 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.

$A$ racing car has a uniform acceleration of $4 \,m s^{-2}$. What distance (in $m$) will it cover in $10 \,s$ after start?

$A$ stone is thrown in a vertically upward direction with a velocity of $5 \, m s^{-1}$. If the acceleration of the stone during its motion is $10 \, m s^{-2}$ in the downward direction,what will be the height attained by the stone and how much time will it take to reach there?

$A$ 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.

An electron moving with a velocity of $5 \times 10^{4} \, ms^{-1}$ enters into a uniform electric field and acquires a uniform acceleration of $10^{4} \, ms^{-2}$ in the direction of its initial motion.
$(i)$ Calculate the time in which the electron would acquire a velocity double of its initial velocity.
$(ii)$ How much distance the electron would cover in this time?