Lookin’ At Slopes: The Calculus of Bad Driving [part 1]

I made a video! Again! It’s just a thing that happens sometimes.

This one is a true story about when I got mad at a taxi driver that skidded through the crosswalk in the rain, and why they were DEFINITELY DOING MATH WRONG.

Part 2 will be out next week, maybe?

Also, obviously I started writing this script a long time ago because it hasn’t rained in ever. But I still think if California focused more of its policies and funding towards higher quality math education, it would make our streets safer.


Today, I was walking across the street and this taxi comes zooming up to the crosswalk at a red light and slams on the brakes and skids across the crosswalk a foot in front of me and I’m like, dude, it’s raining, do you not realize how that affects the derivatives of your position over time, my goodness, I was one foot away from being turned into a pancake by your bad math education.

Now, every driver knows that speed is what happens when you change your position over time (such as the changing position of your car as it approaches the position of the crosswalk) and that acceleration is a change in speed over time (such as, y’know, how even if you change your speed by decelerating as much as your brakes and road conditions allow, you will still be going a positive speed by the time you reach the crosswalk, ). What many drivers don’t seem to know is that these changes are related by mathematical laws. Some drivers are like, you’re here, and then later you’re there? No one can explain this.

And, I can understand that. The idea that anything ever goes anywhere is kind of tricky if you’re Zeno and calculus hasn’t been invented yet. I mean, say you’re 20 meters from the crosswalk. How does one hit pedestrians if before you can get to the crosswalk you have to drive halfway to the crosswalk and then you have to drive halfway between there and the crosswalk and then halfway between there and the crosswalk? and so on. If each of these steps took the same amount of time, then that would be quite an interesting deceleration, you’d never hit anybody like that. But say you drive those first 10 meters in one second, and those next 5 meters in half a second, and the next 2-and-a-half meters in a quarter of a second, it doesn’t matter how many infinite bits of distance you’re adding up, you can break apart those 2 seconds and 20 meters in whatever way you find interesting but 2 seconds later you’ve still gone 20 meters and 2.1 seconds later you’re still trying to ruin my day.

There’s this stereotype about California drivers that whenever it rains, which is rarely enough these days, traffic stops because all the drivers are freaking out like what is this substance all over the ground, we don’t know how to do math to it.

The common wisdom seems to be that when it rains, you should just drive slower. A classic error of calculus, because it’s not really the speed that’s the problem with rain, but how it affects acceleration.

It’s like this: you’re goin’ along at a constant speed, uh, this is time and this is speed and this line is nice and flat so no change in speed is occurring, you’re just driving at 50 miles an hour. But then, oh no, there’s something in front of you, so you slam on the brakes. Now your speed is decreasing, decreasing, until you hit a speed of zero and stop. If you’re at a slower speed to begin with, then this line intersects zero earlier, you can stop faster. So far so obvious.

The slope of this line changes depending on your car and on road conditions: maybe you come to a stop real quickly, or maybe your brakes are bad or the road is icy and you just kinda glide for a while until finally you hit zero. Your car might be able to decelerate real fast when it’s dry, but not so fast when it’s raining, and then even if you start out slower it might take longer to actually stop. You can’t just drive slower, you have to leave more distance between you and the car in front of you, and start braking earlier when you’re coming up on a light, which is why that’s what they tell you in driver’s ed. Whereas if you’ve got lots of room and can decellerate for a long time, you can start at a greater speed even if it takes a while to decelerate to zero. Which is the part they don’t want to tell you in driver’s ed.

Of course, this is graph is kind of misleading because it’s not like the crosswalk is here, this axis shows time, not place. And when we need to stop, we usually don’t care about when to stop so much as where to stop. This graph shows the speed of a taxi that needs to stop at a crosswalk, but let’s overlay the position graph in red, same time axis, different y axis, so we can show where the crosswalk is . Here’s where the Taxi is when I see it coming towards the crosswalk, here’s the crosswalk, 20 meters away. So the driver is going along at a constant speed, that’s this nice linearly increasing distance, realizes it’s a red light and slams the brakes here.

It’s slowing down, and the distance over time starts this nice deceleration curve. Of course, in my case it doesn’t reach the flat zero slope of a stopped car until it’s gone through the crosswalk. Wish it could stop sooner, but once you decide to stop, there’s a max decelleration, you can stop faster if you have better brakes or less momentum or if the ground is dry but there’s always a max slope your speed can drop, which means a max curve your position can take so there we are in the crosswalk. 

Accelleration, speed, and position, these things are related so don’t run me over in the rain. Lookin’ at slopes.

But the story doesn’t end there. We’re leaving off with the taxi driver stopped in the crosswalk but what happens next will surprise you. Or, not really so much, but I wanna talk about hover cars? Anyway see you next time for part 2.