Raise the Hammer
With some high school physics, we can roughly determine a safe vehicle speed given weather conditions for various tires on various surfaces and at various speeds.
By Ted Mitchell
Published February 11, 2009
2,974 is the number of people who died on September 11, 2001. While that's tragic, so are the same number of people who die on U.S. roads every 25.5 days. To the date of this article, this is 106 September 11ths and counting.
Canada has had over 7 September 11ths on our roads since that infamous day. This is the leading cause of death from childhood to middle age, yet even physicians essentially ignore the problem. I see little evidence that our society takes traffic deaths seriously.
What does high school physics have to do with safe driving?
Here in the middle of winter, you can observe very different responses to snow, ice and slush on roads. Some people tend to drive as if nothing much was different, while others crawl down the road.
The police sometimes sum up a crash with the phrase "excessive speed for conditions". This is pretty vague. Can it be quantified?
Yes and no. Things like visibility, driver inattention and poor skill or judgment are not easily analyzed. But every vehicle is subject to the laws of physics.
The first step is to set a benchmark. What is safe? I theorize that adjusting your speed to a constant stopping distance on various surfaces will give a good estimate of what is a "safe speed for conditions".
Say we are talking about a typical urban street with a speed limit of 50 km/h. This being southern Ontario, the enforced speed limit is higher than posted and a speed of 60 km/h is universally tolerated.We need a braking force to slow us down: Ff = ma. On flat ground, that braking force comes from friction Ff = μFn. The coefficient of friction is designated my mu (μ). But Fn = mg, which is the weight of the car. Rearrange, and a = μg. That is, mass cancels, and deceleration is directly proportional to the coefficient of friction.
There are two distinct coefficients of friction for any pair of surfaces: static μs and kinetic μk.
Static means the surfaces do not move relative to each other, while kinetic means they do, and in wheels spinning. Since the static coefficient is always higher than the kinetic coefficient, you should never spin your wheels in snow, ice or any other surface.
Drivers who press on the gas to get out of a slippery situation not only are making things worse for themselves, they really should not have passed high school physics.
But what is μ? It turns out that this is not so easy to determine, and it varies a lot. For dry pavement and good tires, μ is approximately 0.8.
For constant acceleration, v12 - v22 = 2ad. v1 is starting speed, v2 is zero, so rearranging, braking distance db = v12/2μg. Make note of what this means: braking distance is proportional to the square of speed, and directly (inversely) proportional to traction.
For our example above, 60 km/h in real units is 16.7 m/s. So db = 16.72/(2)(0.8)(9.81) = 17.7 m.
This is just the braking distance, for the stopping distance we need to assume a decision-reaction time of about 1 second (relatively quick). So we need to add reaction distance dr = vt = 16.7 (1) = 16.7 m.
Then ds = dr + db.
Stopping distance = 16.7 + 17.7 = 34.4 m.
How much should you slow down to keep constant stopping distance? Because we now have reaction time to deal with, that question becomes a little cumbersome for high school math, so it helps to use a spreadsheet.
| Speed (km/h) | Conditions | Coefficient of friction | Reaction distance (m) | Braking distance (m) | Stopping distance (m) |
|---|---|---|---|---|---|
| 60.0 | dry pvmt | 0.80 | 16.7 | 17.7 | 34.4 |
| 54.3 | wet pvmt | 0.60 | 15.1 | 19.3 | 34.4 |
| 46.7 | hard snow | 0.40 | 13.0 | 21.4 | 34.4 |
| 41.7 | soft snow | 0.30 | 11.6 | 22.8 | 34.4 |
| 35.3 | slushy | 0.20 | 9.8 | 24.5 | 34.4 |
| 26.2 | ice | 0.10 | 7.3 | 27.1 | 34.4 |
Now with snow μ = 0.3, what happens if you don't slow down?
Braking distance is now db = 16.72/(2)(0.3)(9.81) = 47.2m.
Stopping distance is now 16.7 + 47.2 = 63.9m. That is 6.6 car lengths past where you can stop on dry pavement.
| Speed (km/h) | Conditions | Coefficient of friction | Reaction Distance (m) | Braking distance (m) | Stopping distance (m) | Car lengths @ 4.5m |
|---|---|---|---|---|---|---|
| 60.0 | dry pvmt | 0.80 | 16.7 | 17.7 | 34.4 | 7.6 |
| 60.0 | wet pvmt | 0.60 | 16.7 | 23.6 | 40.3 | 8.9 |
| 60.0 | hard snow | 0.40 | 16.7 | 35.4 | 52.1 | 11.6 |
| 60.0 | soft snow | 0.30 | 16.7 | 47.2 | 63.9 | 14.2 |
| 60.0 | slushy | 0.20 | 16.7 | 70.8 | 87.5 | 19.4 |
| 60.0 | ice | 0.10 | 16.7 | 141.6 | 158.2 | 35.2 |
Try out a few numbers and you'll find that the proportional slowing needed for a safe stopping distance is greater with increasing speed.
Several other factors come into play in the real world. Kinetic energy (mv2) decreases markedly at lower speeds, therefore crashes are less severe and less fatal if they happen at lower speeds - and of course, vice versa.
Then there's visibility, traffic congestion, pedestrians, and many important driver factors like inattention and decision - reaction time. Visual processing time (decision speed) is especially sensitive to age-related slowing even in drivers who have normal simple-task reaction times.
Pedestrians and cyclists are at least as affected by slippery conditions (sidewalks not shoveled) and cannot react to avoid collisions as well as usual. They also tend to be hindered by heavy clothing which limits vision and hearing.
Perhaps more important than stopping, control is more likely to be lost in low friction situations, so the vehicle can do more damage and lead to domino effect crashes.
Putting all this together, the prudent driver will treat slippery conditions with respect and slow down at least as much as the table above would suggest.
This analysis is not transferable to traction for acceleration, because it matters if there are two or four wheels driven. Situations exist where friction is marginal for being able to get a two wheel drive vehicle up a small hill or accelerate adequately where four wheel drive has no problem. However, once power is not applied to the wheels, it ceases to matter.
Avoiding a crash requires braking and control and only very rarely calls for acceleration, so it is not surprising that four/all wheel drive has no benefit for safety. Statistics, and a scan of ditches on slippery days, suggest the opposite.
Four wheel drive can be very handy but fails in the safety department. But there is a way to stack the deck in your favour for increased winter traction: snow tires. Not tires with big lugs, those are obsolete. Modern winter tires have a 'mountain snowflake' symbol and several key properties:
Compared with a good set of all-season tires, the coefficient of friction can be up to 25% higher on snow and ice.
Let's say we have a slippery surface with μ = 0.20 for all-season tires and μ = 0.25 for snow tires. Maybe that doesn't seem like much. But you're on the highway traveling 80 km/hr.
| Speed (km/h) | Conditions | Coefficient of friction | Reaction distance (m) | Braking distance (m) | Stopping distance (m) | car lengths @ 4.5m |
|---|---|---|---|---|---|---|
| 80.00 | dry pvmt | 0.80 | 22.2 | 31.5 | 53.7 | 11.9 |
| 80.00 | snow tires | 0.25 | 22.2 | 100.7 | 122.9 | 27.3 |
| 80.00 | all season | 0.20 | 22.2 | 125.8 | 148.1 | 32.9 |
There is a difference in stopping distance of 25.2 m, or 5.6 car lengths.
Put another way, when you have come to a complete stop just in time with your snow tires, the car with all-seasons is still moving - but how fast?
Recall v12 - v22 = 2ad, a = μg. d = db ; braking distance with snow tires, μ = 0.20, and v2 is the unknown we want: the speed of the car without snow tires.
Turns out v2 = 35.7 km/h with all season tires at the point where the car with snow tires has stopped. If there's a stopped transport in front of you that would be bad. Convinced?
The cost of snow tires may seem expensive, but expect a set to last 5 years with moderate use. For most cars, $1500 will buy eight tires and four rims which will last 10 years, and save wear on your summer tires. $150 / year is cheap insurance.
Further reading:
By UrbanRenaissance (registered) | Posted February 12, 2009 at 08:56:27
Great article Ted! This is the kind of example I give any time anyone says "Oh I never really need math/science in my daily life." Most people don't realize that braking distance increases with the square of velocity, so doubling your speed actually quadruples your stopping distance. Incidentally, that equation to derive above is exactly the one police use to determine (roughly) how fast a car was moving based on its skid marks.
Though, the engineer in me does want to clarify your explanation of static vs. kinetic friction. For a rolling wheel (that isn't skidding) the static coefficient is used because at any given instant the part of the wheel touching the ground doesn't move relative to the ground. It briefly touches the ground with only the weight of the car pushing down, that piece of the tire has basically no forward or backwards movement. Once the wheel starts skidding however, the same part of the wheel slides along the ground, meaning in this case, the part of the wheel touching the ground is now moving relative to the ground, implying that the kinetic coefficient should be used.
This is why anti-lock brakes were created, to prevent skidding and decrease stopping distance by taking advantage of the higher coefficient of static friction. Although when its icy out your best protection is a set of snow tires and common sense!
By Ted Mitchell (registered) | Posted February 12, 2009 at 14:41:50
Apologies if the equations don't make much sense, the superscripts and subscripts didn't come through intact. For clarification, it will probably be easier to grab a physics text!
Re kinetic friction, in the box it should have read "as in wheels spinning".
UrbanR's explanation is better.
By Brandon (registered) | Posted February 13, 2009 at 21:25:26
Great article and I mostly agree wholeheartedly, but I have difficulty imagining that tires will last 10 years and be in good condition unless you hardly drive at all.
I also wonder what high performance summer tires will do to your braking numbers on dry pavement?
As far as accelerating to avoid a slipping situation, it's not as cut and dry as you might think. Assuming FWD, if you go into a corner too fast, you'll understeer (vehicle doesn't turn as much as the steering input demands). If you lift sharply, the vehicle will oversteer (vehicle turns more than the steering input demands) due to more traction at the front and less at the rear (weight shifts). If you then accelerate sharply, you can pull yourself out of the skid.
Now, the intelligent driver doesn't deliberately put him/herself in this situation when there are uncontrolled variables, but it can be a lot of fun and good practice when visibility is excellent and there is no one around to be endangered. It's also good to practice it so that should the situation arise unexpectedly you instinctively know how to deal with it.
Sadly, most drivers have no idea what understeer and oversteer mean, never mind how to deal with them when they occur.
By Ted Mitchell (registered) | Posted February 16, 2009 at 12:18:11
8 tires last 10 years. I was trying to save space and let readers do the math.
I don't drive a lot, 10,000 k/year lately, but at that rate the snows last 8-10 years. So 5 years should be easy for average driving, assuming they are only run for 5 months of the winter. A month extra of warm temps can pile on the wear on soft rubber.
"summer tires" - you mean performance tires not rated as all-season? When warm, these will improve stopping, but only by a little, nothing significant like what happens at the other end of the friction scale. Consider that they are usually worse in wet conditions than all-seasons. Also, reaction times would generally overwhelm this small difference that is really only relevant to racers, as it is well beyond what is encountered in normal driving.
e.g. using an extreme case 120 km/h and all-seasons at u=0.8, performance tires at u=0.85, the stopping distance is 4 m shorter, less than a car length.
Oversteer and understeer are very vehicle-specific factors. Otherwise similar cars can have very different control performance at handling limits, and this is certainly influenced by tires etc. I agree that it is useful to have this skill, but as far as safety goes it's nearly irrelevant.
I think most people would agree that these kinds of driving skills are better performed my male drivers, but statistics show that at every age group, men have about twice the crash and fatality rate, drunk or sober, so clearly women, with arguably less advanced driving skills, are the safer drivers. It's all about judgment and risk taking, very little about skill. What matters on the racetrack has very little to do with the road.
By Brandon (registered) | Posted February 16, 2009 at 14:26:15
People who does a lot of research into tires will tend to refer to "all season" tires as "no" season tires, as they're a compromise all around.
High performance summer tires (street tires, not slicks!) will significantly outperform all seasons in wet weather.
That being said, the biggest difference always relates to the gasket between the seat and the wheel.
Here's an interesting couple of videos about braking.
www.tuninglinx.com/videos/fifth-gear-videos/141-fifth-gear-all-things-braking-part1.wmv
www.tuninglinx.com/videos/fifth-gear-videos/142-fifth-gear-all-things-braking-part2.wmv
By Brandon (registered) | Posted February 18, 2009 at 12:22:55
This one is just about tires.
www.tuninglinx.com/videos/fifth-gear-videos/375-fifth-gear-tire-stop-distances.wmv
By Hyppocrate (anonymous) | Posted February 22, 2009 at 09:43:29
Shame on you Ted. With such a doctor shortage in this city you waste your time on this kind of drivel....give your head a shake.
By Ted Mitchell (registered) | Posted March 02, 2009 at 20:44:10
I'm aware of the tradeoff between practicing medicine and pursuing preventive health engineering and political activism. But it takes a special type of person to sit in an office all day listening to lonely people, hearing infinite excuses about why patients can't do this or that which might help them the most, and writing prescriptions and filling out pointless paperwork ad nauseam. Don't get me wrong, I admire those who can do this, but it is can be very draining.
I suppose the reason I spend time on this sort of drivel is that there are some inconvenient facts that society seems to ignore.
Like the fact that motor vehicle crashes are the number one cause of death from childhood to middle age. And that prevention in general is cheaper and more effective than treatment.
So if lawmakers and educators could do their job competently, and the public wouldn't be so resistant to improvement, I would have no reason to add my 2 cents.
By JonathanAquino (anonymous) | Posted December 04, 2010 at 22:29:14
I couldn't get the calculations to work until I realized that the 2 isn't being superscripted properly. So 16.72 should be 16.7^2, etc. In other words, 16.7*16.7.
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