A particle is projected with velocity $v_{0}$ along $x-$ axis. A damping force is acting on the particle which is proportional to the square of the distance from the origin i.e., $ma =-\alpha x ^{2}.$ The distance at which the particle stops:

Read each statement below carefully and state with reasons and examples, if it is true or false 

A particle in one-dimensional motion

$(a)$ with zero speed at an instant may have non-zero acceleration at that instant

$(b)$ with zero speed may have non-zero velocity.

$(c)$ with constant speed must have zero acceleration.

$(d)$ with positive value of acceleration must be speeding up.

Stopping distance of vehicles : When brakes are applied to a moving vehicle, the distance it travels before stopping is called stopping distance. It is an important factor for road safety and depends on the initial velocity $(v_0)$ and the braking capacity, or deceleration, $-a$ that is caused by the braking. Derive an expression for stopping distance of a vehicle in terms of $v_0 $ and $a$.

Given figure shows the $x-t$ plot of one-dimensional motion of a particle. Is it correct to say from the graph that the particle moves in a straight line for $t < 0$ and on a parabolic path for $t >0$? If not, suggest a suitable physical context for this graph.