A train attains a velocity v after starting from rest with a uniform acceleration `alpha`. Then the train travels for sometime withuniform velocity, and at last comes to rest with a uniform retardation `beta`. If the overall displacement is s in time t , show that `t = (s)/(v)+(v)/(2)((1)/alpha+(1)/(beta))`

Explanation

For the accelerated motion,
v = 0+`at_(1)" ""or,"t_(1) = (v)/(alpha)`
and `v^(2) = 0+2alpha s_(1)" ""or,"s_(1) = (v^(2))/(2alpha)`
For the uniform motion, `s_(2) = vt_(2) " ""or," t_(2)= (s_(2))/(v)`
For the retarded motion `0 = v-betat_(3)" ""or," t_(3) = (v)/(beta)`
and `0 = v^(2) - 2betas_(3)" ""or," s_(3) = (v^(2))/(2beta)`
`:. " " s = s_(1)+s_(2)+s_(3) = (v^(2))/(2alpha)+s_(2)+(v^(2))/(2beta)`
or,`" "s_(2)=s-(v^(2))/(2)((1)/(alpha)+(1)/(beta))`
Then `t=t_(1)+t_(2)+t_(3)=(v)/(alpha)+(s_(2))/(v)+(v)/(beta)=(v)/(alpha)+(1)/(v)[s-(v^(2))/(2)((1)/(alpha)+(1)/(beta))]+(v)/(beta)`
`=v((1)/(alpha)+(1)/(beta))+(s)/(v)-(v)/(2)((1)/(alpha)+(1)/(beta))=(s)/(v)+(v)/(2)((1)/(alpha)+(1)/(beta))`