Halliday, Resnick & Walker — Motion Along a Straight Line: Problem 98

Autonomous car passing with oncoming car. Figure 2.37a 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 a long, straight road with one lane in each direction and car C is oncoming at \( v_C = 27.0 \text{ m/s} \) (60 mi/h) at initial distance \( d \) in front of B. Car B is autonomous and is equipped with a computer and radar control system 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 other lane, speed up and pass A, and then pull back into the initial lane to be \( 3.00L \) in front of A and at the initial speed (Fig. 2.37b). The computer control system will allow \( 15.0 \text{ s} \) for the maneuver but only if the front of C will be no closer than \( 10L \) in front of B at the end of the maneuver, as shown in the figure. What is the least value of \( d \) that the system will allow?