A car moves in a straight line, the car accelerates from rest with a constant acceleration `alpha` on a straight road. After gaining a velocity `v`, the car moves with that velocity for sometime. Then car decelerates with a retardation `beta`, If the total distance covered by the car is equal to `s` find the total time of its motion.

Explanation

`V=0+alphat, t_1=v/alpha`
`S_1=1/2alphat_1^2=v^2/(2alpha)`
`O-V=betat_3 , t_3=v/beta`
`O^2-V^2=-2betaS_3`
`S_3=v_2/(2beta)=1/2 betat_3^2`
`t_2=S_2/V=(S-(S_1+S_3))/V=(S-v^2/2(1/alpha+1/beta))/V`
`t=t_1+t_2+t_3`
`=v/alpha+s/v-v/(2alpha)-v/(2beta)+v/beta`
`t=s/v+v/2((alpha+beta)/(alphabeta))`
`s_2=s-(s_1+s_2)=s-(v^2/(2alpha)+v^2/(2alpha))`
`=s-v^2/alpha=s-(vt-s)`
`s_2=2s-vt`
[`S_2` = distance travelled with constant velocity]