Wednesday, December 10, 2014

Lane Change (A Brachistochrone Problem)

Imagine driving your car along the freeway. Initially in the right lane, you decide to get into the left lane. Given that you could lose control of the car if its acceleration vector ever exceeds a given threshold, what is the shortest time in which you can effect the lane change?

(At the end of the maneuver, you must be moving forward at the speed you started with.)

The natural unit of time in this problem is \(\sqrt{\frac{2w}{a_0}}\), where \(w\) is the lane width and \(a_0\) is the acceleration threshold. Expressed in these units, I can make the lane change in \(\sqrt{2}\) ticks of the clock:



During the first half of the motion, the acceleration vector points due left; during the second half, the acceleration vector points due right. The forward velocity component thus remains constant over time.

Formula 1 drivers might maneuver in something like this fashion, but regular drivers probably make a tradeoff between the duration of the maneuver and the difficulty of executing it.

The following graph illustrates some of the difficulty. It shows the component of the acceleration vector along the instantaneous velocity vector (what I call \(a_\parallel\) in my textbook):


Perhaps you can see from the graph what an aggressive sequence this is!

Finally, here is the resulting speed curve:




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