Draw the distance$-$time graph for the following situations

$(a)$ When a body is stationary.

$(b)$ When a body is moving with a uniform speed.

$(c)$ When a body is moving with non$-$uniform speed.

The average time taken by a normal person to react to an emergency is one$-$fifteenth of a second and is called the 'reaction time'. If a bus is moving with a velocity of $60\, km h^{-1}$ and its driver sees a child running across the road, how much distance would. the bus had moved before he could press the brakes ? The reaction time of the people increases when they are intoxicated. How much distance had the bus moved if the reaction time of the driver were $\frac{1}{2}\, s$ under the influence of alcohol ?

Two stones are thrown vertically upwards simultaneously with their initial velocities $u _{1}$ and $u _{2}$ respectively. Prove that the heights reached by them would be in the ratio of $u_{1}^{2}: u_{2}^{2}$ (Assume upward acceleration is $-\,g$ and downward acceleration to be $+g$.

$(a)$ Differentiate between distance and displacement.

$(b)$ Under what conditions is the magnitude of average velocity of an object equal to its average speed ?

Can the distance travelled by a particle be zero when displacement is not zero ?