because they couldn't brake 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: is it even a thing?

Or should we say… negative accelleration?

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 stop light 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 just holding down the brakes real hard.

Deceleration stops when the speed of the car reaches zero, and negative accelleration starts

when you go into reverse, and when you brake while in reverse, slowing down your 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 HOVERCARS, 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 decellerating by

hitting the brakes, because there are no brakes, you'd hit the reverse thrusters that accellerate

you backwards, which would slow you down, I mean negatively care-speed you up to speed

zero but instead of stopping, the hovercar would keep on the reverse thrusters and smoothly

continue the deceleration curve until it were back out of the crosswalk.

And possibly very far away so that I don't come after it on my hoverboard.

Like, here's the regular taxi: vroooom eeeeeee-shnk. mrhhhhshnk.

And the hovercar: vvvvvv pffwhshhh.

Clearly superior, and a much smoother ride.

Good drivers avoid these sorts of nondifferentiable changes in accelleration... y'know, the sudden-change

anglybits.

Although in real life there's lots of little slope-iness going on everywhere in here...

Mm, slopes.

I always found that weird, that you'd get a smoother ride if you kept decellerating

in a hovercar until you're going backwards, than if you just stop.

Which is why to understand what it's like to drive a hovercar I find it helpful to look

at the 2nd derivative, oh, did I mention care-speed is the 1st derivative of position over time

and accelleration is the 2nd?

Calculus.

The art of lookin' 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 are you doin', and he was like,

I am integrating the derivatives of the differential calculus, and it sounded so fancy that they

decided kids just had to memorize all of it, while when someone asked Leibniz what he was

doing, he was probably like, I'm lookin' 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 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 fancy cruise control that lets you directly input a speed, in which

case you can sometimes work on a 1st-derivative level.

And someday we'll all use self-driving cars where you just put in the position you want

to go and 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 hovercars 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 down-wards speed that goes right through the stillness of 0 to continue

backwards, which means your position over time as you approach the crosswalk and back

away again is the perfect smooth curve of a parabola.

And whenever you buy a hovercar 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 the slopes!

It's calculus! (lookin' at slopes).