A car starts from rest and travels with uniform acceleration a, for some time and then with uniform retardation `beta` and comes to rest. If the total time of car is 't', the maximum velocity attained by it is given by

Explanation

`V=0+alpha t, , t_(1)=v/alpha`
`S_(1)=1/2 alpha t_(1)^(2)=v^(2)/(2alpha)`
`O-V=beta t_(3), t_(3)=v/beta`
`O^(2)-V^(2)=-2]`

`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)/(alpha beta))`
`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]