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Spanish train crash: quantifying the acceleration

A tragically motivated homework problem in mechanics

Chances are that you have already seen the dramatic video of the Wednesday Santiago de Compostela derailment. Warning: the following video is brutal.

78 people died and 145 extra ones were injured. In total, it's 223 people – more than the number of passengers, 218 (homework for you: why?). A map allows one to determine that the crash occurred at the top middle point of this Google map.

Using a piece of paper, I estimated the radius of this arc of circle to be \(R\sim 380\,{\rm m}\) or so.

Now, we will need the formula for the centrifugal acceleration\[

a = \frac{v^2}{R}.

\] We will calculate the acceleration for the official speed limit as well as the actual speed.

The official speed limit is \[

v_1=80\,{\rm km/h}\approx 22\,{\rm m/s}

\] while the actual estimated speed was\[

v_1=190\,{\rm km/h}\approx 53\,{\rm m/s}.

\] It's not hard to approximately extract this speed from the video at the top. Google Maps (see the link above) indicates that the distance between the two bridges above the tracks is 250-280 meters and the train made it in 5 seconds or so, between 0:02 and 0:07 of the video. Divide 265 meters by 5 seconds and you get 53 meters per second.

Recall that the speed gets squared when we compute the acceleration. Because the driver exceeded the maximum allowed speed \(2.375\) times, the maximum centrifugal acceleration was surpassed by the factor of \(5.64025\) (all this precision is bogus, of course: it's for you to accurately verify your calculations). That's quite a factor. At any rate, the two accelerations are – using the simple formula\[

a_1 \approx 1.3\,{\rm m/s}^2,\quad a_2 \approx 7.3\,{\rm m/s}^2.

\] While the first, allowed one is about \(1/7\) of \(g\), the second one is about \(3/4\) of the Earth's attractive gravitational acceleration. And that makes a difference.

The train is subject to \(9.8\,{\rm m/s}^2\) of the vertical acceleration and \(1.3\) or \(7.3\,{\rm m/s}^2\) of horizontal acceleration. The angles (away from the vertical axis) determining the direction of the total centrifugal-plus-gravitational acceleration (the "total gravity" from the passengers' viewpoint in the sense of general relativity) obey \[

\tan\alpha_{1,2} = \frac{a_{1,2}}{g}

\] and they are \[

\alpha_1\approx 0.13\,{\rm rad}\approx 7.5^\circ \text{ and }\alpha_2\approx 0.64\,{\rm rad}\approx 36.7^\circ

\] for the speed limit and the actual speed, respectively. The first angle is modest; the second, actual angle is stunning. Even if the tracks were optimized (non-horizontal) for the recommended speed limit, \(80\,{\rm km/h}\), the direction of the total acceleration during the actual ride of death would still be almost \(30^\circ\) away from the vertical direction.

Should it be enough for derailment? Well, experimentally speaking, it was enough.

Theoretically, it's useful to imagine that the direction of the total acceleration as the vertical one; the actual "down the train" direction deviates from it by those \(36.5^\circ\).

In the most naive model, if the cross section of the train were a square, the center-of-mass were in the middle of the square, and the wheels were at the extreme left-and-right endpoints of the square, then the critical angle would be \(45^\circ\). In reality, the train is a "slightly tall" rectangle and the wheels are "somewhat closer to each other". Both of these deviations from the simplest model make the overturning more likely i.e. they reduce the critical angle. The wheel flanges are pushing in the opposite direction and make the train somewhat more stable in similar situations but it wasn't enough. I don't know what was the height of the center of mass of the wagons. There are many subtler points in derailment that you may learn e.g. from Wikipedia.

At any rate, I wonder whether the driver was calculating the angle of the total acceleration before he or she tried whether \(190\,{\rm km/h}\) is an OK speed for that curve. He or she should have. I am saying "he or she" to fight against the stereotype that killers are male, and to fight against the underrepresentation of women among killers. I hope that the Feminazis will praise me for that. It seems to me that the Spanish bureaucrats spend much more time by overwhelming self-employed babes with impenetrable paperwork than by verifying a remotely acceptable speed of the trains. ;-)

Don't forget about the "conical wheels" explanation by Feynman why trains don't need a differential, why they don't get derailed in curves under normal circumstances (low enough speeds), and why the flanges aren't the heart of the right answer.

BTW if you want to see that Czech kids are better engine drivers than Spanish adults, see this 2-minute 1960 video on the Pioneer Railway in front of the Pilsner zoo that was fully operated by kids between 1959 and 1976. The adults only donated the trains to the kids and they decorated the kids by the communist symbols. My father (who was living just 200 meters away from the tracks) was already building capitalism as a kid – during the very construction, he was taking some iron/tracks from the Pioneer Railway and selling it as a raw material. ;-)

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reader Dimension10 (Abhimanyu PS) said...

Nice mix of all the topics you cover ( Except that you took string theory to the non-stringy low-energy classical non-relativistic limit : ) )

reader Dilaton said...

Huh, has the driver survived?

Was it really an accident, maybe the driver has done the caclulations ... :-/ ?

reader Gordon said...

In Feynman clips, it is great seeing the joy he gets in
explaining how things work. If everyone had his child-like curiosity and delight in understanding, the
world would be a much better place.

reader Gene Day said...

Thirteen days ago we came breathtakingly close to an even greater tragedy here at the San Francisco Airport when a Boeing 777 dropped far below the required landing speed, stalled and hit the seawall. The passenger compartment survived the impact except for an opening in the pressure bulkhead at the rear, through which three people were cast when the plane spun rapidly to the left before stopping. These three were killed out of a total of 306 passengers and crew members. Had the final airspeed been just a bit lower we would have lost around 300 souls.
Unfortunately, pilots and train engineers are human and humans occasionally lapse into a state in which they are unconscious of their surroundings. The one thing you don’t do as a pilot is fail to maintain airspeed; you just don’t do that, but three professional pilots on the flight deck of Asiana Fl. 214 did it simultaneously. Of course the one thing that train operators don’t do is enter a curve too fast. Not ever.

reader Luboš Motl said...

Dear Dilaton, yes, he survived. In fact, before the crash, he did a simplified calculation and made a telephone call informing others that he is at 190 km/h and he would be going to derail:

He will be investigated.

reader it does not matter said...

Hummm, I must say this is quite a piece of thinking, kudos for that, because although it's something any kid about to leave primary school should know how to calculate, the thing here is who thinks of doing so, and you have.

But, also, I must say that you're basing your estimations on that google map and... that would be wrong. I live some 40 something miles away from that place and I usually go by that point (the highway is used to avoid going through Santiago de Compostela) twice a week. In fact, I had been there less than two hours before. But I'll explain myself.

- The image that google maps has is actually quite old, several years. I would say something around three to four years old without looking at the copyright of the images.

- That railway we see, a "one lane" railway, is the old one to go from Santiago to Ourense. Today, there are three "lanes", two for the AVE, one for the "classic" train. That's something that can be seen in the video.

- So, the AVE train actually comes in an almost straight line from the tunnel being constructed (at the moment of the image) down and to the right of the highway bridge.

- The "classic" railway has been "pushed" inwards, so to speak, so the actual radius for its turn is probably smaller, but, at the same time, the railway for the AVE is not exactly on that same path, so it has been kind of "lightened", I mean, probably made bigger comparing the image from what I have seen over and over so many times.

This all is to say that, in my honest opinion, I do believe the radius is somewhat greater than what can be thought because of this outdated image.

About the matter of "optimized" railways, yes, their cross section is clearly at an angle, but it's certainly not that much. You have to take into account that that point is some 3.5 kms away from the station, so, whatever the radius of the turn, the train should never go at any kind of high speed since then it has to stop in a very short space. Ant it is supposed to do so in a comfortable manner for the passengers.

reader Gene Day said...

It surely will be determined that multiple failures contributed to the Spanish catastrophe, just as for the San Francisco Airport incident on the 13th. When automated systems fail it is up to human operators to be attentive. That’s why they are there.
It should be noted that air and rail travel is remarkably, if not perfectly, safe. For the eight years following 2001 there were zero fatalities in the US due to commercial air travel despite many millions of operations. I’m sure that rail travel in Europe can boast similar statistics.

reader Luboš Motl said...

Dearl Gene, air and rail travel is surely not "perfectly safe". Just two weeks ago, train derailed near Paris and about 6 fatalities were there:

Air travel tragedies are almost on daily order. I would say that they're still major contributors to the death of people. We experienced a not-so-negligible number of stories. Two years ago, the whole professional Russian ice-hockey team Yaroslavl - including its Czech players- was exterminated during an air crash:

Just a year earlier, the Polish president and 100 top representatives of the Polish elite was killed in air crash in Smolensk, Russia

I could give you comparable stories about railway traffic. And you know, 9/11 is fun to be cut from the story but it was really a big story, too.

My understanding was that the Spanish engine driver had full access to control the train but he just screwed it and it was the only reason.

reader Gene Day said...

Actually, Lubos, I didn’t cut 9/11 from the story at all.
The 2001-2009 accident-free interval in the US aviation began with the crash of American Airlines Fl. 587, which occurred in New York City two months after 9/11, on November 12, 2001. The crash of this Airbus A300 killed 260 people when the rudder detached from the aircraft due to misapplication. The safe interval ended with the crash of Colgan Air Fl. 3407 on February 12, 2009 in Buffalo, New York, due to insufficient airspeed. There were no survivors of either crash.
Obviously, nothing is perfectly safe but my point was that air travel in the US is astonishingly safe, considering that you are moving in a thin-walled aluminum tube at barely sub-sonic speeds. The probability of a passenger being killed is only a bit over 10^-8 per flight and that’s just amazing! Other countries have worse records but pilot error is almost always the cause of air disasters.
My other point is that the operator is supposed to be paying attention. Of course the guy in Spain wasn’t.

reader Gene Day said...

It’s surprising to me that this train did not have an automatic braking system. Even my personal car automatically applies the brakes if it senses an impending collision. In the GPS age surely the train’s computer knew that the speed had to be reduced. Did it just sound a warning that the operator ignored?

reader Luboš Motl said...

Dear Gene, I am in no disagreement about facts etc., still, I may react a bit differently emotionally because I know some flight fatalities around.

10^{-8} per flight is a very small number, indeed. In fact, it's smaller even when counted "per unit time". A human lives for 700,000 hours or so, so the probability of natural death is like 1 part per million per hour. So even when an average flight takes several hours, when several is multiplied by 10^{-8} - assuming that your figure is right - one is below the natural background for the same period of time.

Whoops, I am an idiot. One must multiply the airline tally by the average number of passengers - then they're of the same order. So during the flight, you about double your rate of death risk. They're comparable. Am I right?

reader johnl said...

In California, this would have been impossible. A passenger train going 2 mph over the speed limit is stopped and forced to recycle.

reader Chris said...

The map shows that this curve was one of a series of sharp curves in the track that they had already gone through. They could have derailed on any one, and the drivers must have been well aware of what was likely to happen.

reader RandomReader said...

Nice way of thinking.
How did you calculate curve radius? I read in a newspaper that curve radius was "slightly above 1/2km" and my hand calculations on paper give a 640m radius curve. Just wondering, because that would result in 60% of your calculated acceleration.
Also, the photo on google maps is old. You can see the high speed line works there, and if you go to the map view you can see the high speed line, about 6km straight, coming out of a tunnel.

reader Doktor Bob said...

Hi Lubos,
Nice write up, thanks for sharing your skills.
Love / DB

reader Luboš Motl said...

Thanks - and fun picture. ;-)

reader Luboš Motl said...

Using a finger of mine, I "pinned" a piece of paper to a center of a circle and watched the other side of the circle going almost exactly along the arc under consideration. In this way, I determined the center of the circle in this way and on the paper, I indicated the distance between the center and the tracks, and measured the distance between these two points using the scale in the legend, left lower corner of Google Maps.

I may try to do a more accurate measurement, including an estimate of the error margin, later. Perhaps, I could measure it in a digital way using Mathematica etc.

reader Luboš Motl said...

Update: I got almost exactly 400 meters radius, 399 meters if you wish, via Mathematica.

Via ColorDetector that also shows coordinates of the cursor, I measured the coordinates x,y of three points - in the middle of the tracks: under two bridges and in the middle of the path.

Then I solved a pair of algebraic equations for the point D that is equally far from the three points above, and calculated the shared distance from them. In my setup, it was 455 pixels radius. Then I measured 100 meters from the legend to be 585-471=114 pixels, and 455/114 times 100 meters gave me almost exactly 400 meters.

Mathematica code:

a = {855., 326.};
b = {996., 343.};
c = {1137., 411.};

Solve[EuclideanDistance[a, {d1, d2}] ==
EuclideanDistance[b, {d1, d2}] &&
EuclideanDistance[b, {d1, d2}] ==
EuclideanDistance[c, {d1, d2}], {d1, d2}]

EuclideanDistance[a, {871.668, 780.99}]

unit = 585 - 471


reader Tuxedo said...

Hm, I think your calculation is perfectly fine up to the point where you declare yourself an idiot. The number 10^{-8} is the expected number of fatalities divided by the number of passengerflights, where e.g. a flight with 200 passengers counts as 200 passengerflights. So the number is the expected probability that any given passenger should die as consequence of a crash, and not the probability that the plane will crash (which is slightly higer, since not all crashes has a 100% fatality rate.)

If all passengers always died in a crash, the two probabilities would be equal: For if we lost one flight out of one million with 200 passengers average, there would be 200 / 200 million fatalities per passenger flight, i.e. 1 to 1 million.

reader Doktor Bob said...

Oh, were working very hard not just on technology but also on entertaining people, like you! Why, because people wants to be entertained right :)

You write in a funny way and thus makes it easier to learn.

I would recommend you to keep your eyes open on our NI-Week demonstration 5-8 aug in Texas.

I predict epic planetary victory.

Kind Regards / DB
Information and technology can solve all our problems.

reader Luboš Motl said...

Right. Not sure where your violent words come from because that's exactly the calculation I meant at the very end, with the very same result. Doubling the rate of death during the flight - 1 in 700,000 or approximately 1 in million gets enhanced to 2 in million - is exactly what I described. Yes, I assume that all the flight passengers die.

reader Tuxedo said...

What I meant whith that comment is that, as far as I can see, you should not multiply the airline tally with the average number of passengers, as you do in the last paragraph, since that's already there in the 10^{-8} number. I'm very sorry if you found my words violent, that was very far from my intention. I was simply alluding to the fact that you wrote "Whoops, I am an idiot" when in fact you were not :-)

reader Luboš Motl said...

OK, but if 10^{-8} is the chance for a single passenger to die in an airflight, then it is 100 times smaller than the chance that an average person dies for other reasons during that time, isn't it?

10^{-8} is one-per-100-million while the chance to die within an hour etc. is 1-in-million, right?

reader Tuxedo said...

Excactly. So your statement "So even when an average flight takes several hours, when several is
multiplied by 10^{-8} - assuming that your figure is right - one is
below the natural background for the same period of time." is entirely correct (it was the later correction where you multiplied with average number of passengers per flight, and thus upped the probability, I had issues with.)

And by the way, I don't think you being an idiot is a "subtle" question, I think it's pretty well demonstrated within five sigmas to be "out of" the question :-)

reader Gene Day said...

The 10^-{8} risk of death per flight is, of course, just an order of magnitude estimate. Many things can increase it but, in at least one case, it actually seems too large already.
Southwest Airlines has carried well over one billion passengers and all but one survived their flight (provided one ignores those few who doubtless expired of natural causes in mid-flight).
The one was a young guy who was killed by other passengers when he stormed the flight deck many years ago.
I usually fly Southwest and figure that I am ten times more likely to be killed by the effects of another Chixulub type meteorite than by a commercial aircraft accident.

reader Gene Day said...

Yep, nothing subtle about that!

reader Fer137 said...

What derailed, was the heavy diesel generator wagon, with a very high center of mass. (an invention of politician for the train out of electrified and non-electrified roads)
(in spanish)

In the video you can see (the wagon behind the locomotive derailed. The other end, heavy wagon, crushes.)

reader Jon said...

I think a more meaningfull gauge would be as for cars. Fatalities pr million or billion person kilometer. Then you can compare easier?

reader Luboš Motl said...

Clever method, John!

reader spencershawn said...

Very insightful. Thank you very much that is interesting how that all could happen. I hope they had railroad liability insurance or else it could get messy.

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