5 ms·

The Mathematics Behind Stopping a Car

10 points by prakash 17y ago

presidentender 17y ago
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 17y ago
  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 17y ago
    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 17y ago
      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 17y ago
        Wind resistance is V^3. Drag is V^2, but you multiply that by the distance traveled.
  - mschy 17y ago
    If I slam on my brakes for an emergency (animal in road, accident ahead, etc.), I can stop from 60mph in approximately 110-120 feet. That's not that far from a magically appearing brick wall. At 60mph, you'll cover the whole of that distance in approximately 1.25 seconds. Assuming even a few tenths of a second to notice that I'm stopping, and a few more tenths to realize that I'm stopping FAST, and those "large distances" start to look quite reasonable. Particularly if your car can't brake like a sports car, or if your mind sometimes drifts.
  - nradov 17y ago
    I wrecked my car several years ago on CA Hwy 17 due to a blocked road. Although I wasn't speeding, when I came around a blind curve there was another car that had spun out and stopped completely blocking my lane. It was impossible to completely avoid. When the Highway Patrol officer arrived I asked him if he was going to write a report. He told me that he would if I wanted him to, but if he did then he'd have to cite me for colliding with a stopped vehicle. Even though it wasn't really my fault, that's how the law is written. Fortunately the insurance investigation determined that I wasn't at fault.
ars 17y ago
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 17y ago
  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.

10 points by prakash 17y ago

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 17y ago
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 17y ago
  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 17y ago
    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 17y ago
      Wind resistance is V^3. Drag is V^2, but you multiply that by the distance traveled.
- mschy 17y ago
  If I slam on my brakes for an emergency (animal in road, accident ahead, etc.), I can stop from 60mph in approximately 110-120 feet. That's not that far from a magically appearing brick wall. At 60mph, you'll cover the whole of that distance in approximately 1.25 seconds. Assuming even a few tenths of a second to notice that I'm stopping, and a few more tenths to realize that I'm stopping FAST, and those "large distances" start to look quite reasonable. Particularly if your car can't brake like a sports car, or if your mind sometimes drifts.
- nradov 17y ago
  I wrecked my car several years ago on CA Hwy 17 due to a blocked road. Although I wasn't speeding, when I came around a blind curve there was another car that had spun out and stopped completely blocking my lane. It was impossible to completely avoid. When the Highway Patrol officer arrived I asked him if he was going to write a report. He told me that he would if I wanted him to, but if he did then he'd have to cite me for colliding with a stopped vehicle. Even though it wasn't really my fault, that's how the law is written. Fortunately the insurance investigation determined that I wasn't at fault.

ars 17y ago

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 17y ago
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.