 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 skips 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 you know how even if you change your speed by decelerating as much as your brakes and road conditions allow, you'll still be going a positive speed over 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 you're there and no one can explain this. And I can understand that. The idea that anything ever goes anywhere is kind of tricky. If you're Xeno 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 two 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 a problem with rain, but how it affects acceleration. It's like this. You're going along at a constant speed. 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 50 miles an hour. But then, oh no, something's 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 stop really quickly. Or maybe your brakes are bad or the road is icy and you just kind of 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. That's why that's what they tell you in Driver's Ed, whereas if you've got lots of room and can decelerate for a long time, you can start at a greater speed even if it takes a while to decelerate to zero, which is a part they don't want to tell you in Driver's Ed. Of course, this 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 from 20 meters away. Here's the crosswalk. 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 have been sooner, but once you decide to stop, there's a max deceleration. 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. Acceleration, speed, and position, these things are related, so don't run me over in the rain looking 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 really not so much, but I want to talk about hover cars. Anyway, see you next time for Part 2.