In coverage of the tragic Philadelphia train crash, the news and public are amazed that the train’s speed is over twice of the limit for the curve being rounded. The speed was 106 mph, and the speed limit was 50 mph.
The key physics phenomena involved, however, go like the square of the speed or velocity, and are a factor of 4.5 times greater than they would be at the speed limit!
The force needed to be supplied by the train tracks to hold the train on the curve is
F = m v^2 / r , where m is the mass of the part of the train being turned at that time, v is the velocity and speed, and r is the circular radius of that section of track. So while the velocity was 2.12 times the speed limit, the square of the velocity and the centripetal force was 4.49 or about 4.5 times that which would be necessary at the velocity of the speed limit to hold the train on the curve.
This 4.5 factor in the force would act to dislocate the railroad tracks at the bend. What acts sooner is the fact that the railroad tracks are applied at the bottom of the train, and not opposite its center of mass. So they act as a tripping step, and the train tips over instead.
The kinetic energy of the entire train is proportional to the train’s velocity squared.
K. E. = 1/2 M v^2 , where M is now the mass of the entire train, and v the velocity of the train.
Here again, the kinetic energy is 4.5 times that which the train would have had if it were obeying the speed limit. All of that energy will be dissipated as the train comes to a stop. The energy would be absorbed by train cars sliding on the ground and running into each other, by deformations of the cars, and by objects and passengers slamming into the train.
Prevention of Accidents
Apparently, the fact that Amtrak is losing money and Congress is cutting still more is slowing the automatic train stopping system, even in the most traveled Eastern corridor.
Its odd that I have a $100 GPS monitor in my car that turns the speed indicator red every time I exceed the speed limit for whatever road I am on. Connecting that to an alarm would awaken a drowsy operator, or catch the attention of one being distracted by anything else. Hooked to a smart phone or on a smart phone, it could automatically call train dispatchers. This could remove some incidents of train crashes.
It probably the budget of Amtrak, but every airplane carrying 250 people has to have a co-pilot. Until the track braking system is installed, this should be instituted in trains carrying lots of passengers.