
The Mathematics Behind Stopping a Car - prakash
http://arachnoid.com/lutusp/auto.html
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presidentender
In driver's ed, they teach a 2-second following distance from the car in front
of you (4 seconds, perhaps, if possible; in traffic, however, 4 seconds is a
hole to be filled by a greedy-algorithm-piloted car). The arguments in favor
of this are that it's easy, and it scales linearly with velocity, so that
you'll be further behind when moving faster, and have more time to react.

At first blush, this article seems to contradict that: since stopping time is
O(n^2), we should need a quadratic stopping distance formula. However, upon
further consideration, it seems to me that the 2-second rule ought to be just
fine. That allows for reaction time (assumed to be 1.5sec in the article), and
the quadratic part is taken care of by the fact that the other car can't just
stop dead in front of us, but has to take the same quadratic stopping time
that we do.

~~~
lutorm
Yes, if you were to drive as if every car in front of you could run into a
magically appearing brick wall, then you would really need such large
distances. Luckily, that's not the case.

However, if you want to drive in compliance with the "Basic Speed Law", which
states "No person shall drive a vehicle upon a highway at a speed greater than
is reasonable or prudent having due regard for weather, visibility, the
traffic on, and the surface and width of, the highway, and in no event at a
speed which endangers the safety of persons or property", ie. taking into
account that you can run into stationary things, then you really have to
conclude that your _visibility_ must scale as v^2. So if you are going around
a corner and need to be able to stop in case the road is blocked (ie CA 17)
then this _is_ the distance you care about.

As an aside, I think it's quite an interesting question as to what the optimal
distance is if you take into account the fact that people will cut into your
cushion space and make it <50% of what it was. It seems it will be the largest
distance that other drivers still think is too small to merge into.

~~~
Retric
It's not V^2 due to reaction time. Assuming 1.5 seconds it's around 1.5s * V +
k * V^2. And using his numbers:

    
    
      10MPH = 27feet
      20MPH = 64feet
      80MPH = 496feet 
    

The point is even at low speeds you still have a fairly long stopping
distance. Which is why you can see so many fender benders in stop and go
traffic.

PS: At high speeds wind resistance can dramatically affect stopping distance,
but your breaks have limits on how much heat they can take so it's a fairly
convoluted equation.

~~~
lutorm
High-velocity behavior is v^2, which where it matters. The linear part quickly
becomes negligible. And wind resistance is still v^2 so it'll still go as v^2
only slower.

The brake limit is an interesting one, though. Apparently brakes are not
providing a constant brake _torque_ as one would naively expect, but instead a
constant dissipation power. This is why everyone can lock their brakes going
slowly, but it's normally quite hard to do if you're going 80mph. (For a
normal "family" car.)

~~~
Retric
Wind resistance is V^3. Drag is V^2, but you multiply that by the distance
traveled.

------
ars
I think at non slow speeds, most people have about the same stopping distance.

This is because the faster you are going the harder you typically press on the
brake.

Very very few times will you need an emergency stop and max out your brakes.

Most of the time you do gentle braking, and a slightly harder braking at
greater speeds.

The net result is that typical (not minimum) braking distances do not change
much with speed.

~~~
rlpb
Yes - the article assumes that brakes operate by converting the kinetic energy
of the car at a constant rate and thus this is the limiting factor.

Yet with decent brakes most of us do an emergency stop as hard as possible
without skidding, so the limiting factor is the traction with the road.

