A truck has to carry a load in the shortest time from one point to another at a distance L from the first.It can only start up or slow down at the same acceleration a. What maximum velocity must the truck attain to satisfy this condition ?

Options

A. `sqrt(La)`
B. `sqrt(2La)`
C. `sqrt(3La)`
D. `sqrt(5La)`

Answer

A

Explanation

Let v be the maximum velocity attained and t the total time of journey. t. is the duration of acceleration and retardation. Then `v= 0 + a t.`
`:. L = (1)/(2) a t.^(2) + v (t-2t.) + (1)/(2) "at."^(2)`
`=a ((v)/(a))^(2) + v (t- (2v)/(a)) = (v^(2))/(a) + vt - (2v^(2))/(a) = vt - (v^(2))/(a)`
`t= (L)/(v) + (v)/(a) rArr (dt)/(dv) = - (L)/(v^(2)) + (1)/(a)` When t is minimum, `(dt)/(dv) = 0 :. v_("max") = sqrt(La)`