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Two cars, \( A \) and \( B \), are traveling in the same direction with velocities \( v_{a} \) and \( v_{b} \) respectively. When car \( A \) is at a distance \( d \) behind \( \operatorname{car} B \), the brakes on \( A \) are applied, causing a deceleration at a rate \( a \). Show that to prevent a collision between \( A \) and \( B \) it is necessary that :
\[
v_{A}-v_{B}=\sqrt{2 a d}
\]
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