Adversarial Ambulation

When crossing the street, I typically work on the assumption that drivers at minimum are unaware of me, and possibly seeking to hit me. I've often wondered whether crossing perpendicular to the road is the best strategy, or whether angling my path away from an oncoming car would be better. If we imagine crossing a road of width w, while a car approaches starting y0 up the road, we can write the distance squared (to avoid taking a square root) as

where t is the time since entering the road, the vs are the respective velocities of the pedestrian and the car, and θ is the angle from perpendicular for the path. We want to find the angle that gives us the greatest distance (or distance squared) when the car is at its closest point in time. To find that, first we find the time of the closest approach:

Now, we could plug this into the first equation, and try to maximize over θ, but things are already getting ugly, so instead, let's plug in some values and plot it. For a 20 ft wide road, with a car starting 100 ft away and driving at 30 mph, we can plot the closest approach for different angles and walking speeds:

At high enough speeds, it is worthwhile to angle yourself away from the car, but unless you want to book it, straight across is probably best. You may notice the slowest speed in blue puts you farther than the next highest in orange – This is because it's so slow, the car passes before you get significantly into the road. Definitely avoid that orange speed though – That kink at around 38° is a distance of 0 ft, otherwise known as being run over!