Total Recall of Bad Mathematics

You know there’s something wrong with a movie if during the climatic action scene in the movie you are busy thinking about the bad science in this fiction movie. The movie was the 2012 remake of Total Recall. The original movie was full of its own bad science about life on Mars. You could forgive it because it was an Arnold movie on Mars. So what if they forgot how gravity works. In the remake they have neglected the laws of physics again. Let’s give them the magnetic cars and just focus on the big elevator that runs between Great Britain and Australia. That’s not much of a spoiler since it is introduced in the opening scroll and is a major plot device with little regard for reality.In the movie our characters board the elevator each morning in Australia and fall through the earths core to arrive in the UK 17 minutes later. There is another plot time that reaffirms how long this takes. There is no visible method of propulsion so we can assume the method of propulsion is gravity itself. In the movie gravity is a constant force from the surface of the planet down to the core where it is instantaneously suspended before being reversed on the other side of the core moments later. You get that? The occupants on the elevator are strapped into chairs that look like a ride at Six Flags.

The first thing that bothered me about this was the time. I was unsure while sitting in the darkened theater if I remembered the radius of the Earth. I was pretty sure the circumference at the equator was 25,000 miles. So by dividing by 2π I should get the radius. Let’s call it 4,000 miles to keep the math simple. I was not about to light up my phone-based calculator in the theater to check my math but I thought that was pretty close. That means the distance traveled was about 8,000 miles in just 17 minutes or roughly 28,000 miles per hour averagespeed. Starting from zero that is some pretty wicked acceleration. But we already said that the elevator was in free fall or 1G of acceleration. More on that later. What complicates this further is that in those 17 minutes we have to both accelerate and decelerate.

Let’s just focus on the 4,000 mile trip in 8.5 minutes. Starting at zero that means that the acceleration force would be about 10G⁽¹⁾ or ten times the normal pull of gravity on Earth. So if a person weighed 100 pounds they would have to support 1,000 pounds of weight on their sitting frame. According to a reliable source, the Internet, “most people black out at 5-6 Gs. 8 is too much and 12 is defenitely lethal”⁽²⁾. In a roller coaster you get 4-5 Gs for a few seconds and even that can be fatal. This would not be a good plan for your regular commute. So after 8.5 minutes of this 10G acceleration we reach the peak speed of 107,795 mph. Mind you that 17,000 mph is escape velocity. I really hope this is all taking place in a vacuum so that the air in front of the elevator doesn’t turn to plasm and destroy the whole machine. What? Did Colin just open a window???? So much for the vacuum theory. Now he’s climbing outside. I’m starting to think this might not be realistic.

So assuming you could build such an elevator how would it work. First of all let’s use gravity for our primary acceleration. It there anyway. If we assume that gravity is constant all the way to the core before mysteriously flipping then the math is quite easy⁽¹⁾: 1G of acceleration gets you up to 11,187 mph after just 8.5 minutes. It’s not quite escape velocity but you have traveled almost 800 miles straight down. You manage to get to the core in about 19 minutes and you’re traveling about 25,135 mph. It is at this point when according to the movie, gravity flips, and you can use gravity to slow you down until you eventually come to a full stop. This would be sort of like the parabola that a ball thrown into the air travels but on the dimension of gravity.

Using movie physics what would the ride be like inside the elevator? For anyone who has ridden the parachute drop at Six Flags knows the experience would be one of weightlessness. As the elevator falls ever faster and faster towards the core the riders would experience weightlessness right from the beginning as they fall at the same rate. Once they pass through the core and are being slowed by gravity they would experience the 2Gs of gravity exerting on them. One of those Gs would slow their speed until they eventually came to a stop. Since this is movie physics in a miraculous vacuum we will have to assume there’s no loss of energy due to friction. Or we’ll have to assume that they have magnets in the tube to overcome the friction.

The problem with the 1G acceleration model is that gravity would decrease as you approached the core based on your distance from the center of what has now become two masses. Take the Earth and draw a plane slicing through the earth perpendicular to the elevator shaft. Weigh both pieces. Calculate the center of mass both halves. Your local gravity would be calculated by calculating the gravitational pull of the two parts of the earth, subtracting the one behind you from the one in front of you as you fell. So gravity would decrease from 9.8m/s² to 0. This also means that your acceleration toward the core would slow as you fall.

The movie could fix all of the problems by inventing artificial gravity, putting the elevator in a vacuum, and lining the entire tube with magnets for propulsion. This would allow them to accelerate to 10G without burning up the Earth and artificially create a period of zero G at the midpoint of the time. Problems solved. But then they couldn’t have gone outside to take a look around.

I think I have torn the physics of this movie up enough but I have one last question. Near the end of the movie our hero parks a helicopter on the roof of the elevator. When they come out the other side the helicopter is still on the roof. Figure that one out. They made a big point in the movie of showing that the cabins with the seats rotated in place.

None of this has anything to do with the quality of the movie. It was ok if you ignore the laws of physics.

(1) http://www.cthreepo.com/lab/math1.shtml
(2) http://forum.ebaumsworld.com/archive/index.php/t-73659.html