Question

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 $B$ is at a distance $d$ ahead of the car $A$, the driver of the car $A$ applied the brake producing a uniform retardation $a$. There will be no collision if

• A. $d<\frac{(v_1-v_2{)}^2}{2a}$
• B. $d<\frac{(v_1^2-v_2^2)}{2a}$
• C. $d>\frac{(v_1-v_2{)}^2}{2a}$
• D. $d>\frac{(v_1^2-v_2^2)}{2a}$

Answer 1

The correct option is C $d>\frac{(v_1-v_2{)}^2}{2a}$

initial relative velocity of car A w.r.t. car B
$u_{rel}=v_1-v_2$
Final relative velocity of car A w.r.t. car B
$v_{rel}=0$
as there should be no collision, both cars will finally move with same velocity
$a_{rel}=-a$
$\Rightarrow S_{rel}=\frac{v_{rel}^2-u_{rel}^2}{2a}=\frac{(v_1-v_2{)}^2}{2a}$
To avoid collision $d\ge S_{rel}\Rightarrow d\ge \frac{(v_1-v_2{)}^2}{2a}$