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(C) Let the relative velocity of car $A$ with respect to car $B$ be $u_{rel} = v_1 - v_2$.Since car $A$ is decelerating at $a$ and car $B$ is moving w

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$d > \frac{(v_1 - v_2)^2}{2a}$

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(C) Let the relative velocity of car $A$ with respect to car $B$ be $u_{rel} = v_1 - v_2$.Since car $A$ is decelerating at $a$ and car $B$ is moving with constant velocity $v_2$,the relative acceleration is $a_{rel} = -a$.For the collision to be avoided,the relative displacement $s$ of car $A$ with respect to car $B$ before it comes to rest relative to $B$ must be less than the initial distance $d$.Using the equation of motion $v_{rel}^2 = u_{rel}^2 + 2a_{rel}s$,where $v_{rel} = 0$ at the point of minimum separation:$0 = (v_1 - v_2)^2 - 2as$$s = \frac{(v_1 - v_2)^2}{2a}$To avoid collision,the initial distance $d$ must be greater than the relative stopping distance $s$.Therefore,$d > \frac{(v_1 - v_2)^2}{2a}$.

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

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