Before discussing the physics (oh, dear) I want to make clear that this film is excellent. A truly fantastic sequel to J.J. Abrams’ Star Trek reboot. Indeed one of my favourite Trekkie shows/films. There are amusing, as well as heartfelt, scenes between Kirk (Chris Pine), Spock (Zachary Quinto) and Uhura (Zoe Saldana) and then there’s the ever-majestic Benedict Cumberbatch as the uber-villan Khan (spoiler alert).
So, great cast, tight script, superb directing and effects are stunning (you really have to see this in IMAX 3D to do it justice). Where does it go wrong? Well, simply put, the physics is dreadful. To such a degree that I think it’s intentionally doing it to annoy physics majors. There are too many to mention, but let’s focus on just two main scenes.
The opening scene has Kirk leading alien villagers away from a volcano that may explode. Never mind that it is stated that this super-volcano will destroy the planet and hence the audience should wonder what the point of leading villagers away from it has to do with anything. The solution it appears is to have Spock descend in a shuttle, on a line, and manually detonate a cold fusion bomb to freeze the volcano. I’m really not sure where to begin with this, everything from this having to be done manually, to the idea that the volcano can be frozen… Just too bizarre to argue about as a cold fusion bomb would be a nuclear bomb without the compression heating (i.e. it doesn’t cool stuff). So that’s weird but then it gets worse.
Heroically (albeit pointlessly) Kirk leads the villagers away and jumps off a cliff into the sea (no idea how he survives a drop of what looks like hundreds of metres but possible I guess) and then swims to the Enterprise..! Yes that’s right, the spaceship is under the water. For some reason it has to hide there (not that it could be seen in space by the primitive aliens).
A big issue is that spaceships are designed to withstand one atmosphere of pressure (since they are holding back Earth-like atmosphere against the vacuum of space). Here the Enterprise needs to withstand the weight/pressure of the water around it, which is significantly larger than a single atmosphere as any submariner or scuba diver will tell you. In fact pressure increases at one atmosphere for every 10 metres deeper you go.
If we assume the Enterprise is 70 metre tall (an exhaustive analysis of the dimensions are discussed here) the bottom of the ship would have to withstand seven times the pressure it would actually have been built for. Even taking into account safety margins in engineering the Federation Flagship it seems a little too much to ask of the poor ship.
But wait, there’s more. The ship rises majestically into the sky (how do space-based impulse drives work under water?) pushing an enormous volume of seawater out of the way. In particular, the torqueing action of so much water on the 70m-radius saucer section would snap the fragile neck of the Enterprise. There is a reason submarines are streamlined!
How much torque? Assuming the saucer is barely submerged it has one atmosphere of pressure (101325 Newton/metre^2) which means that the force on the saucer section is Pi * 70^2 times this, or nearly 1.56 billion Newtons. Now we assume the lever arm of the resultant torqueing action is the radius of the saucer (it’s actually more as the saucer sits forward but is inclined which lessons the torque but this order of magnitude). This means another factor of 70, so the torque is 0.1 trillion Newton metres. *Snap*
So assuming that the cold fusion bomb *arggggghhhh* works, and that the Enterprise can fly out into space without breaking its neck, the movie proceeds and we fast forward to the culmination of bad physics in the final scene.
The Enterprise is orbiting near the Moon when it is crippled in an attack (spoiler alert too late?). It then begins to fall to Earth, in minutes. Considering it isn’t initially moving while, say, the Apollo moon missions were, and they took days it’s hard to see the sense of urgency in repairing the ship. Certainly enough time to don a radiation suit in any case (definitely too late for a spoiler alert).
Ignoring the timescale then, the weirdest thing is that the Enterprise crew experiences the direction of gravity changing as the ship ‘plummets’ to Earth. Since the ship is in free-fall they would actually experience a far more serene weightlessness like the International Space Station experiences (orbit is just continual free-fall). Far from being flung into one floor/ceiling/wall surface after another the crew would get to happily float around and play with spherical water-blobs like Cmdr Chris Hadfield (in fact, drowning inside a floating water blob would be a real hazard with the amount of water we see inside all the tubes in engineering!)
So in conclusion, ignoring warp travel, transporters (which don’t work inside volcanoes apparently) and various other weird bits of Trekkie lore this film was stupendously bad in terms of physics. Yet I loved it so I can happily say I’m a fan first and scientist second.