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.
