$A$ particle moves for $8 \, s$. It first accelerates from rest and then retards to rest. If the retardation is $3 \, \times$ the acceleration, then the time for which it accelerates is: (in $s$)

The $v^2$ versus $s$ graph of a particle moving in a straight line is shown in the figure. From the graph,some conclusions are drawn. State which statement is wrong?

Consider a vehicle moving with a velocity $54 \,km/h$. At a distance of $400 \,m$ from the traffic light, brakes are applied. The acceleration of the vehicle after the application of brakes is $-0.3 \,m/s^2$. The vehicle's position relative to the traffic light is: (in $\,m$)

The velocity of a particle moving in the positive direction of the $x$-axis varies as $v = 5 \sqrt{x}$. Assuming that at $t = 0$,the particle was at $x = 0$,what is the acceleration of the particle in $m/s^2$?

What is the relation between displacement,time,and acceleration for a body undergoing uniform acceleration?

Derive the equations of motion for constant acceleration using the method of calculus.