Minimum time of travel of an electric car- M.C.Q. from one dimensional motion

An experimental electric car can produce a maximum acceleration of 2m/s². Its brakes can produce a maximum retardation of 8m/s². What is the minimum time required by this electric car to start from rest and come to a stop, covering a distance of 500m?

(a) 10s (b) 15s (c) 20s (d) 25s (e) 30s

This question may appear to be a bit difficult at the first sight, but it is a simple one which can be solved using the basic equations of motion. The maximum velocity ‘v’ attained during the motion is given by

v² = 0+2a₁s₁ during the accelerated motion and

0 = v²-2a₂s₂ during the decelerated motion. From these equations, s₁ = v²/2a₁ and s₂ = v²/2a₂

Here s₁+ s_{2 =} 500m, a₁= 2m/s² and a₂ = 8m/s².

Therefore, 500 = v²/2a₁ + v²/2a_2. On substituting for a₁ and a_2, v = 40m/s.

But we have, v=0+a₁t₁ and 0=v-a₂t₂ for the accelerated and decelerated parts of the motion respectively so that t₁=v/a₁=40/2 = 20 s and t₂=v/a₂ = 40/8 = 5 s.

The total minimum time therefore is 25 s, given in option (d).