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. ;-)
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. ;-)
Spanish train crash: quantifying the acceleration
Reviewed by DAL
on
July 25, 2013
Rating:
Reviewed by DAL
on
July 25, 2013
Rating:
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