Halliday, Resnick & Walker — Motion Along a Straight Line: Problem 96

Autonomous car passing with following car. Figure 2.36a gives an overhead view of three cars with the same length \( L = 4.50 \text{ m} \). Cars A and B are moving at \( v_A = 22.0 \text{ m/s} \) (49 mi/h) along the right-hand lane of a long, straight road with two lanes in each direction, and car C is moving along the passing lane at \( v_C = 27.0 \text{ m/s} \) (60 mi/h) at initial distance \( d \) behind B. Car B is autonomous and is equipped with a computer control system using radar to detect the speeds and distances of the other two cars. At time \( t = 0 \), the front of car B is \( 3.00L \) behind the rear of car A, which is at \( x_{A1} = 0 \) on the x axis. We want B to pull into the passing lane, speed up and pass A, and then pull back into the right-hand lane, \( 3.00L \) in front of A and at the initial speed. The computer control system will allow \( 15.0 \text{ s} \) for the maneuver and only if the front of C will be no closer than \( 3.00L \) behind A at the end of the maneuver, as in Fig. 2.36b. What is the least value of \( d \) that the system will allow?