Two cars A and B are travelling in the same direction with velocities $v_1$  and $v_2$ $(v_1>v_2).$When the car A is at a distance 'd' ahead of the car B, the driver of the car A applies the brake producing a uniform retardation 'a' there will be no collision when:

Given that the velocities are $V_1 and V_2$ and car 'A' is 'd' distance ahead of car 'B'.

Hence, initial relative velocity, u = $V_1 - V_2$

Final relative velocity = 0

From equations of motion, $v^2 = u^2 - 2as (as the car is retarding)$

$v^2 - u^2 = - 2as$

Hence, v = 0

$0^2 - {(V_1 - V_2)}^2 = - 2as$

 $\frac{{\left(V_1 - V_2\right)}^2}{2a} = s$

To avoid collision 's' should be less than 'd'

$d > s$

Therefore,  $d > \frac{{\left(V_1 - V_2\right)}^2}{2a}$