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

$(a)$ Derive second equation of motion $S=u t+\frac{1}{2} a t^{2}$ graphically where the symbols have their usual meanings.

$(b)$ A car accelerates uniformly from $18\, km h ^{-1}$ to $36\, km h^{-1}$ in $5$ seconds. Calculate the acceleration and the distance covered by the car in that time.

In a long distance race, the athletes were expected to take four rounds of the track such that the line of finish was same as the line of start. Suppose the length of the track was $200\, m$.

$(a)$ What is the total distance to be covered by the athletes ?

$(b)$ What is the displacement of the athletes when they touch the finish line ?

$(c)$ Is the motion of the athletes uniform or nonuniform ?

$(d)$ Is the displacement of an athlete and the distance moved by him at the end of the race equal ?

Ali while driving to school computes the average speed for his trip to be $20\, km h^{-1}$. On his return trip along the same route there is less traffic and the average speed is $30\, km h^{-1} .$ What is the average speed for Ali's trip ?

Ravi told his friend that his house is $1 \,km$ toward south from the main Post Office. Express displacement and the distance moved by the friend from the Post Office when he arrives at Ravi's house. Mention the reference point chosen by you.