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(C) To avoid a collision,the relative displacement of car $A$ with respect to car $B$ must be less than or equal to the initial separation $d$ when th

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$d > 25$

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(C) To avoid a collision,the relative displacement of car $A$ with respect to car $B$ must be less than or equal to the initial separation $d$ when the relative velocity becomes zero.Let the velocities be $v_A = 30 \, m/s$ and $v_B = 20 \, m/s$.The relative initial velocity is $u_{rel} = v_A - v_B = 30 - 20 = 10 \, m/s$.The relative acceleration is $a_{rel} = a_A - a_B = -2 - 0 = -2 \, m/s^2$.For no collision,the relative velocity must become zero before or at the moment the displacement equals $d$.Using the equation $v^2 = u^2 + 2as$ for relative motion:$0^2 = (10)^2 + 2(-2)d_{rel}$$0 = 100 - 4d_{rel}$$4d_{rel} = 100$$d_{rel} = 25 \, m$.Thus,for no collision,the initial distance $d$ must be greater than $25 \, m$.

Two cars $A$ and $B$ are moving in the same direction with velocities $30 \, m/s$ and $20 \, m/s$. When car $A$ is at a distance $d$ behind car $B$,the driver of car $A$ applies brakes,producing a uniform retardation of $2 \, m/s^2$. There will be no collision when ......... $m$.

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