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:

## No comments:

Post a Comment