calculate

How To Calculate Maximum Car Speed + Examples (Mercedes C-180, Bugatti Veyron)

How do you determine the maximum possible speed your car can go? Well, one rather straight-forward option is to just get into your car, go on the Autobahn and push down the pedal until the needle stops moving. The problem with this option is that there’s not always an Autobahn nearby. So we need to find another way.

Luckily, physics can help us out here. You probably know that whenever a body is moving at constant speed, there must be a balance of forces in play. The force that is aiming to accelerate the object is exactly balanced by the force that wants to decelerate it. Our first job is to find out what forces we are dealing with.

Obvious candidates for the retarding forces are ground friction and air resistance. However, in our case looking at the latter is sufficient since at high speeds, air resistance becomes the dominating factor. This makes things considerably easier for us. So how can we calculate air resistance?

To compute air resistance we need to know several inputs. One of these is the air density D (in kg/m³), which at sea level has the value D = 1.25 kg/m³. We also need to know the projected area A (in m²) of the car, which is just the product of width times height. Of course there’s also the dependence on the velocity v (in m/s) relative to the air. The formula for the drag force is:

F = 0.5 · c · D · A · v²

with c (dimensionless) being the drag coefficient. This is the one quantity in this formula that is tough to determine. You probably don’t know this value for your car and there’s a good chance you will never find it out even if you try. In general, you want to have this value as low as possible.

On ecomodder.com you can find a table of drag coefficients for many common modern car models. Excluding prototype models, the drag coefficient in this list ranges between c = 0.25 for the Honda Insight to c = 0.58 for the Jeep Wrangler TJ Soft Top. The average value is c = 0.33. In first approximation you can estimate your car’s drag coefficient by placing it in this range depending on how streamlined it looks compared to the average car.

With the equation: power equals force times speed, we can use the above formula to find out how much power (in W) we need to provide to counter the air resistance at a certain speed:

P = F · v = 0.5 · c · D · A · v³

Of course we can also reverse this equation. Given that our car is able to provide a certain amount of power P, this is the maximum speed v we can achieve:

v = ( 2 · P / (c · D · A) )1/3

From the formula we can see that the top speed grows with the third root of the car’s power, meaning that when we increase the power eightfold, the maximum speed doubles. So even a slight increase in top speed has to be bought with a significant increase in energy output.

Note the we have to input the power in the standard physical unit watt rather than the often used unit horsepower. Luckily the conversion is very easy, just multiply horsepower with 746 to get to watt.

Let’s put the formula to the test.

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I drive a ten year old Mercedes C180 Compressor. According the Mercedes-Benz homepage, its drag coefficient is c = 0.29 and its power P = 143 HP ≈ 106,680 W. Its width and height is w = 1.77 m and h = 1.45 m respectively. What is the maximum possible speed?

First we need the projected area of the car:

A = 1.77 m · 1.45 m ≈ 2.57 m²

Now we can use the formula:

v = ( 2 · 106,680 / (0.29 · 1.25 · 2.57) )1/3

v ≈ 61.2 m/s ≈ 220.3 km/h ≈ 136.6 mph

From my experience on the Autobahn, this seems to be very realistic. You can reach 200 Km/h quite well, but the acceleration is already noticeably lower at this point.

If you ever get the chance to visit Germany, make sure to rent a ridiculously fast sports car (you can rent a Porsche 911 Carrera for as little as 200 $ per day) and find a nice section on the Autobahn with unlimited speed. But remember: unless you’re overtaking, always use the right lane. The left lanes are reserved for overtaking. Never overtake on the right side, nobody will expect you there. And make sure to check the rear-view mirror often. You might think you’re going fast, but there’s always someone going even faster. Let them pass. Last but not least, stay focused and keep your eyes on the road. Traffic jams can appear out of nowhere and you don’t want to end up in the back of a truck at these speeds.

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The fastest production car at the present time is the Bugatti Veyron Super Sport. Is has a drag coefficient of c = 0.35, width w = 2 m, height h = 1.19 m and power P = 1200 HP = 895,200 W. Let’s calculate its maximum possible speed:

v = ( 2 · 895,200 / (0.35 · 1.25 · 2 · 1.19) )1/3

v ≈ 119.8 m/s ≈ 431.3 km/h ≈ 267.4 mph

Does this seem unreasonably high? It does. But the car has actually been recorded going 431 Km/h, so we are right on target. If you’d like to purchase this car, make sure you have 4,000,000 $ in your bank account.

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This was an excerpt from the ebook More Great Formulas Explained.

Check out my BEST OF for more interesting physics articles.

Sources:

http://ecomodder.com/wiki/index.php/Vehicle_Coefficient_of_Drag_List

http://www.mercedes-benz.de/content/germany/mpc/mpc_germany_website/de/home_mpc/passengercars/home/_used_cars/technical_data.0006.html

http://www.carfolio.com/specifications/models/car/?car=218999

How To Calculate the Elo-Rating (including Examples)

In sports, most notably in chess, baseball and basketball, the Elo-rating system is used to rank players. The rating is also helpful in deducing win probabilities (see my blog post Elo-Rating and Win Probability for more details on that). Suppose two players or teams with the current ratings r(1) and r(2) compete in a match. What will be their updated rating r'(1) and r'(2) after said match? Let’s do this step by step, first in general terms and then in a numerical example.

The first step is to compute the transformed rating for each player or team:

R(1) = 10r(1)/400

R(2) = 10r(2)/400

This is just to simplify the further computations. In the second step we calculate the expected score for each player:

E(1) = R(1) / (R(1) + R(2))

E(2) = R(2) / (R(1) + R(2))

Now we wait for the match to finish and set the actual score in the third step:

S(1) = 1 if player 1 wins / 0.5 if draw / 0 if player 2 wins

S(2) = 0 if player 1 wins / 0.5 if draw / 1 if player 2 wins

Now we can put it all together and in a fourth step find out the updated Elo-rating for each player:

r'(1) = r(1) + K * (S(1) – E(1))

r'(2) = r(2) + K * (S(2) – E(2))

What about the K that suddenly popped up? This is called the K-factor and basically a measure of how strong a match will impact the players’ ratings. If you set K too low the ratings will hardly be impacted by the matches and very stable ratings (too stable) will occur. On the other hand, if you set it too high, the ratings will fluctuate wildly according to the current performance. Different organizations use different K-factors, there’s no universally accepted value. In chess the ICC uses a value of K = 32. Other approaches can be found here.

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Now let’s do an example. We’ll adopt the value K = 32. Two chess players rated r(1) = 2400 and r(2) = 2000 (so player 2 is the underdog) compete in a single match. What will be the resulting rating if player 1 wins as expected? Let’s see. Here are the transformed ratings:

R(1) = 102400/400 = 1.000.000

R(2) = 102000/400 = 100.000

Onto the expected score for each player:

E(1) = 1.000.000 / (1.000.000 + 100.000) = 0.91

E(2) = 100.000 / (1.000.000 + 100.000) = 0.09

This is the actual score if player 1 wins:

S(1) = 1

S(2) = 0

Now we find out the updated Elo-rating:

r'(1) = 2400 + 32 * (1 – 0.91) = 2403

r'(2) = 2000 + 32 * (0 – 0.09) = 1997

Wow, that’s boring, the rating hardly changed. But this makes sense. By player 1 winning, both players performed according to their ratings. So no need for any significant changes.

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What if player 2 won instead? Well, we don’t need to recalculate the transformed ratings and expected scores, these remain the same. However, this is now the actual score for the match:

S(1) = 0

S(2) = 1

Now onto the updated Elo-rating:

r'(1) = 2400 + 32 * (0 – 0.91) = 2371

r'(2) = 2000 + 32 * (1 – 0.09) = 2029

This time the rating changed much more strongly.

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The Standard Error – What it is and how it’s used

I smoke electronic cigarettes and recently I wanted to find out how much nicotine liquid I consume per day. I noted the used amount on five consecutive days:

3 ml, 3.4 ml, 7.2 ml, 3.7 ml, 4.3 ml

So how much do I use per day? Well, our best guess is to do the average, that is, sum all the amounts and divide by the number of measurements:

(3 ml + 3.4 ml + 7.2 ml + 3.7 ml + 4.3 ml) / 5 = 4.3 ml

Most people would stop here. However, there’s one very important piece of information missing: how accurate is that result? Surely an average value of 4.3 ml computed from 100 measurements is much more reliable than the same average computed from 5 measurements. Here’s where the standard error comes in and thanks to the internet, calculating it couldn’t be easier. You can type in the measurements here to get the standard error:

http://www.miniwebtool.com/standard-error-calculator/

It tells us that the standard error (of the mean, to be pedantically precise) of my five measurements is SEM = 0.75. This number is extremely useful because there’s a rule in statistics that states that with a 95 % probability, the true average lies within two standard errors of the computed average. For us this means that there’s a 95 % chance, which you could call beyond reasonable doubt, that the true average of my daily liquid consumption lies in this intervall:

4.3 ml ± 1.5 ml

or between 2.8 and 5.8 ml. So the computed average is not very accurate. Note that as long as the standard deviation remains more or less constant as further measurements come in, the standard error is inversely proportional to the square root of the number of measurements. In simpler terms: If you quadruple the number of measurements, the size of the error interval halves. With 20 instead of only 5 measurements, we should be able to archieve plus/minus 0.75 accuracy.

So when you have an average value to report, be sure to include the error intervall. Your result is much more informative this way and with the help of the online calculator as well as the above rule, computing it is quick and painless. It took me less than a minute.

A more detailed explanation of the average value, standard deviation and standard error (yes, the latter two are not the same thing) can be found in chapter 7 of my Kindle ebook Statistical Snacks (this was not an excerpt).