 So in my last video I ranted about a taxi driver who almost hit me in the crosswalk because they couldn't break fast enough in the rain, and I was thinking about this problem in a few ways. They couldn't stop before the crosswalk, but really it's that they couldn't decrease their speed fast enough to get from 50 to 0 before hitting the crosswalk. Or maybe it's really that the slope of their decreasing speed, out of all possible slopes, was not a steep enough slope. Too little deceleration. Now some people after seeing my last video protested my use of the term deceleration. Deceleration isn't even a thing, or should we say negative acceleration because the opposite of speeding up isn't slowing down, it's negative speeding up. And that's the problem with cars. They have separate modes for forward and reverse. When you're coming up to a stoplight, you don't slow down by putting the car into reverse, you hit the brakes. And after you slow to a stop using the brakes, if you want to start going backwards back out of the crosswalk, it's not like you do it by holding down the brakes real hard. Deceleration stops when the speed of the car reaches zero, and negative acceleration starts when you go into reverse. And when you brake while in reverse, slowing down, you're slowing down, that's negative deceleration, which is also known as acceleration. Yep, hitting the brakes while in reverse actually speeds you up in the positive direction because really we're talking about velocity, not speed. Velocity, it's speed that cares about direction, care speed. But if we had hover cars, there's no frictional attachment to the relative zero position of the earth, no friction based brakes that care about things like whether you're stopped or not. In a hover car, you'd be zooming up to the crosswalk and instead of decelerating by hitting the brakes, because there are no brakes, you'd hit the reverse thrusters that accelerate you backwards, which would slow you down, I mean negatively care speed you up to the speed zero. But instead of stopping, the hover car would keep on the reverse thrusters and smoothly continue the deceleration curve until it were back out of the crosswalk, and possibly continue until it's very far away so that I don't come after it on my hover board, like here's a regular taxi. And then the hover car. Clearly superior and a much smoother ride. Good drivers avoid these sorts of non-differentiable changes in acceleration, you know the sudden change angly bits, although in real life there's lots of these little sloppinesses going on everywhere in here. Slopes. I always found that weird that you get a smoother ride if you keep decelerating in a hover car until you're going backwards then if you just stop. Which is why to understand what it's like to drive a hover car I find it helpful to look at the second derivative. Oh did I mention care speed is the first derivative of position over time and acceleration is the second derivative? Calculus the art of looking at slopes. Side note, I'm pretty sure the reason Newton gets the credit for inventing calculus instead of Leibniz was that someone was like hey Newton what you doing? And he was like I am integrating the derivatives of the differential calculus and it sounded so fancy that they decided that kids just had to memorize all of it. While when someone asked Leibniz what he was doing he was probably like I'm looking at slopes. I like slopes. Do you like slopes? I like slopes. Anyway that's how it goes in my head but it's the second derivative that I care about because when you're driving that's what you have control over. You can accelerate and decelerate using gas and brakes. That's it unless you've got some fancy cruise control that lets you directly input a speed. In which case you can sometimes work on a first derivative level and someday we'll all be using self-driving cars where you just put in the position you want to go and then they calculate the rest. Ah technology, lowering derivatives for the common good. Although if we were going to work directly with position rather than acceleration I'd prefer straight up teleportation but if you like differentiable modes of transportation then hover cars are the mathematically more beautiful choice of vehicle with which to narrowly avoid hitting pedestrians. Just look at that smooth flat deceleration that creates a constant sloping downward speed that goes right through the stillness of zero to continue backwards which means your position over time as you approach the crosswalk and back away again is a perfect smooth curve of a parabola. And whenever you buy a hover car you should always check the range of your second derivative because more powerful thrusters means steeper slopes for your first derivative and tighter parabolas. Parabola It's got all the slopes. Look at all the slopes. It's calculus. Looking at slopes.