Unstoppable is a 2010 disaster film somewhat inspired by a real life story of a runaway freight train and the attempts to stop it. It involves a number of action-filled scenes that contain key mathematical concepts, including a potentially catastrophic derailment on a turn.
Here’s the trailer:
Math Scenario 1: Assessing the Damage
In the movie, the runaway train has 8 cars full of toxic, flammable molten phenol, which is used to create plastics.
If each car contains a cylindrical tank that measures 6.1 metres long with a diameter of 2.44 metres, how many litres of phenol is being carried by the train?
Total volume phenol = volume of 1 tank * #tanks
Total volume phenol = PI * r^2 * L * #tanks
Total volume phenol = PI * 1.22^2 *6.1 * 8
Total volume phenol = 228.2 m^3
There are 1000 litres per cubic metre, so there are 228200 litres of phenol stored in the train’s tanks.
Math Scenario 2: A Missile the Size of the Chrylser Building
In one of the memorable scenes early in the movie, a character describes the train as a “missile the size of the Chrysler building”.
We can work out the approximate volume of the Chrysler building using the height and cross-section area of the building.
The building doesn’t have the same cross-sectional area all the way up, so we can estimate that the building’s average cross-sectional area is half the base area. The base area according to Google is 7689 m^2. The height is about 282 m (not including the spire).
Building volume = height * average cross sectional area
Building volume = 282 * 0.5 * 7689
Building volume = 1084149 m^3
The published volume of the somewhat larger Empire State Building is about 1 million cubic metres as well, so our estimate is probably in the right range.
Let’s say the average volume of a rail carriage is about twice the one we calculated for the phenol tank, to take into account other structure making up the carriage. We can work out how many carriages are equivalent to the Chrysler building:
# carriages equivalent = volume of building / volume per carriage
# carriages equivalent = 1084149 / (2 * 228.2)
# carriages equivalent = 2375 carriages
In the movie, the train clearly doesn’t have anywhere close to that number of carriages – so even if the average carriage volume is larger than we estimated, the movie quote is still exaggerating.
We can also compare the building and the train by weight.
The Empire State Building is a bit bigger than the Chrysler building, and has a weight of 365000 tonnes. Let’s say the Chrylser building is a bit smaller at around 200000 tonnes.
The heaviest train in history was less than 100000 tonnes, and it was more than 7 km long – much bigger than the train in the movie.
So whether by weight, or by volume, the comparison to the Chrysler building is a big exaggeration, but it’s an action movie so that’s okay 🙂
Math Scenario 3: Chasing Down the Train
At one stage in the movie they go after the train with a helicopter.
If the helicopter has a top speed of 240 km/hr, the train has a 30 km head start on the train, and the train is travelling at 120 km/hr, how much further will the train have travelled by the time the helicopter catches up?
Relative to where the helicopter starts, we can write equations for the positions of the train and helicopter along the track after t hours:
helicopter = 240 * t
train = 30 + 120 * t
The helicopter will catch up when its position is the same as the train. Therefore we can equate the two equations:
helicopter = train
240 * t = 30 + 120 * t
120t = 30
t = 0.25 hrs
The helicopter will take a quarter of an hour to catch up, but how far will the train have travelled during this time?
Train travel = speed * time
Train travel = 120 * 0.25
Train travel = 30 km
The train will have travelled an additional 30 km by the time the helicopter catches up.
Math Scenario 4: Safe Cornering Speed
The slower they can get the train, the less likely it is to derail at the final curve in the movie.
Let’s say the old curve in the movie has a maximum cornering speed of 50 km/hr without risk of derailment.
The train is currently travelling at 120 km/hr. The drivers manage to start slowing the runaway train with another train engine, decelerating the train by 20 km/hr per minute.
We can write an expression for the speed of the train after time t (with t given in minutes):
Train speed = 120 – 20t
We can also rewrite this with t given in hours:
Train speed = 120 – 1200t
We can then calculate how long the train will take to slow to 50 km/hr:
Time to slow = (current speed – desired speed) / deceleration
Time to slow = (120-50) / 20 km/hr/minute
Time to slow = 3.5 minutes
To calculate how far the train will travel during the time it takes to slow from 120 to 50 km/hr, we need to integrate this expression with respect to t:
Integral of train speed = 120t – 600t^2
And evaluate its value between t = 0 and t = 3.5 minutes. First however we need to convert these times into units of hours to be consistent:
t = 0 and t = 3.5/60 = 0.05833
Now calculate the distance:
Distance = [120t – 600t^2 for t = 0.05833] – [120t – 600t^2 for t = 0.0]
Distance = 4.958 – 0
Distance = 4.958 km
The train will travel almost 5 km during the time it takes to slow from 120 to 50 km/hr.
Real Life Example – CSX 8888 incident and Runaway Ramps
The movie is somewhat based on a real-life example where a 47 car train drove itself without human supervision for two hours in Ohio in the USA.
In that real-life example, a crew caught the runaway train and coupled to its rear car.