# Finding the velocity $v$ for which the total time the object moves from its initial position to the position where it stops is minimal

A body first moves with a constant velocity $v$ along a track with a length of $L=5 \ [m]$ and then decelerates with an acceleration $a=2 \ [\frac{m}{s^2}]$ till it stops. How to find the velocity $v$ for which the total time the object moves from its initial position to the position where it stops is minimal?

Let $v$ be the starting velocity. The time it takes to move along a track with constant velocity is $$t_1 = \frac{L}{v} = \frac{5}{v}$$
The time it takes to decelerate to zero velocity $$t_2 = \frac{v-v_0}{a} = \frac{v}{2}$$
The total time we have to minimize is $$t(v) = \frac{5}{v} + \frac{v}{2}$$