Killing Time With Recreational Math - Star Trek Into Darkness
In the movie 'Star Trek - Into Darkness' the starship Enterprise is attacked by the USS Vengeance and comes out of warp between the Earth and the Moon. In the scene when they come out of warp the pair of ships appear to be fairly close to the Moon.
The Enterprise is then disabled and it falls to the Earth. It seems to take about 30 minutes or less for it to hit the Earth's atmosphere and heat up due to the re-entry.
Is this realistic? In this post, I do the math for you.
Note: if you hate math then just skip down to the results section and read over the answer. It will surprise you.
The Spreadsheet
In the calculation below I work out the time for the Enterprise to fall from a height of 100,000 km above the surface of the Earth.
The screenshot above is the spreadsheet I have used to calculate the time for the Enterprise to fall from a height above the Earth and hit the atmosphere. In this calculation I assume that it had zero velocity when it started falling. To explain the calculation I will go through each cell one by one.
Cell B1: this is the universal gravitational constant 6.67 x 10-11m3/kgs2.
Column A: In column A the spreadsheet simply keeps track of the time in seconds. To limit the size of the spreadsheet I increment the time by 10 seconds in each subsequent row. You can vary the increment size as you see fit.
Column B: In column B the spreadsheet keeps track of the time in minutes. To do this, it simply divides the value of the cell to the immediate left by 60.
Column C: In column C the spreadsheet keeps track of the time in hours. To do this, it simply divides the value of the cell to the immediate left by 60.
Cell D6: This cell calculates the distance of the Enterprise from the centre of the Earth. The average radius of the Earth is 6378 km and it is multiplied by 1000 to get km. In this example the Enterprise starts falling 100,000 km above the surface of the Earth so that too is multiplied by 100,000 km.
Cell D7: This cell calculates the height decrement each time step. Column I calculates the distance fallen at each time increment and so this cell is a simple calculation of the initial height minus the distance fallen so far.
Column E: Column D calculates the height from the centre of the Earth but Column E calculates the height above the surface of the Earth. It is just the value in Column D minus the radius of the Earth in metres. This is not used in any calculations in the spreadsheet, it is just useful for knowing when the ship hits the surface of the Earth.
Column F: This column is just the mass of the Earth. In retrospect I should have stored this value in a single cell but oh well, it doesn't really matter in the grand scheme of things.
Column G: This column calculates the acceleration due to gravity which is a = GM/r2. The value for G is in cell B$1 (I used an absolute reference for the row indicator). The value of M is the mass of the Earth and it is in column F. The value for r is the distance from the centre of the Earth and it is in column D.
You will notice how small the acceleration due to gravity is at 100,000 km. On the surface of the Earth it is about 9.8 m/s2 but at 100,000 km it is only 0.035 m/s2. This means that the Enterprise will fall really slowly at the start.
There is a deliberate mistake
The mistake in the equation above is that I am ignoring the pull of gravity from the Moon. The Moon will actually pull it in the other direction and the Enterprise will actually fall slightly more slowly than this calculation. Maybe I will add in this improvement in another post.
Column H:
Cell H6 starts off the velocity at zero.
In cells H7 and further down the velocity of the Enterprise at each time step is calculated using the kinematic formula:
v = v0 + a (tprev - tnow)
where,
v is the new velocity,
v0 is the velocity in the previous time step and
a is the gravitational acceleration at that height
tprev - tnow calculates the time increment.
Column I: This is the workhorse of the calculation. It calculates the distance fallen by the Enterprise at each time step using the kinematic formula:
d = d0 + v (tprev - tnow) + 1/2 a (tprev - tnow)2
The Results
So after all of that, the spreadsheet calculates that the Enterprise will strike the Earth in about 60,570 seconds or 16.8 hours.
That's a long time and it is a lot different than the 20 to 30 minute fall time in the movie. Kirk and the Enterprise actually had a long time to fix their warp core. Also you will notice that the Enterprise would be falling at about 10.7 km/s. This means that it would have torn through the atmosphere in 3 to 5 seconds before impacting on the ground.
Okay, so maybe the Enterprise was closer than 100,000 km from the Earth when it started falling. The beauty of a calculation like this is that it can be easily adjusted to work out different scenarios and the results are in the table below.
| Height | Fall Time |
|---|---|
| 100,000 km | 65,570 seconds = 16.8 hours |
| 50,000 km | 23,120 seconds = 6.4 hours |
| 25,000 km | 9,350 seconds = 2.6 hours |
| 10,000 km | 3,245 seconds = 54 minutes |
| 5,000 km | 1,658 seconds = 28 minutes |
| 400 km | 300 seconds = 5.0 minutes |
Closing Words
It turns out that the Enterprise would have to be about 5,000 km above the surface of the Earth when it was disabled by the Vengeance if it were to hit the atmosphere in about 30 minutes.
For curiousity's sake I had the spreadsheet work out the time to fall from the height of the orbit of the International Space Station which is 400 km above the surface of the Earth.
It turns out to be only 5 minutes but of course this object would need to have zero forward velocity for this to happen. Maybe Jeff Bezos' New Glenn rocket will do just that, take people straight up and have them fall from that height back down for a total weightless duration of nearly 10 minutes.
Thank you for reading my post.
Other Posts In My Recreational Math Series
- Testing the excel random function.
- Make Your Own Bell Curve
- Strange Attractors.
- Let's Travel To Alpha Centauri
- Calculate Sunset and Sunrise Times
- Conway's Game of Life.
- Caffeine Half-Life.
- Let's Simulate a Radioactive Sample
- Journey Through the Centre of Psyche
Post Sources
https://en.wikipedia.org/wiki/Star_Trek_Into_Darkness
https://en.wikipedia.org/wiki/Gravitational_constant
https://en.wikipedia.org/wiki/New_Glenn
https://en.wikipedia.org/wiki/Equations_of_motion
Nice! You should do a movie-debunked video on this. No way it was only 5,000km when the Moon is 384,400km away; there's no camera angle to fake that, not the way it looks.
So much is done to enhance the feel of space films, sometimes bending the laws of physics is the best cinematic option.
Yeah, they had about 16 hours or more to fix their engines. Lots of time for a rescue or at least a tow from Star Fleet as well.
Most impressive good sir! Your arithmetic prowess is not to be trifled with.. Now, where’s a good place to get some Dilithium Crystals around here!?
That's classified.
Your logic is impeccable. Are you related to Spock?
I wish. My 23andMe results showed English, Irish and general north European ancestry, no Vulcan DNA whatsoever.
Interesting and fun, as always.
While I am not sure that I saw that episode, I would be happy to offer consultancy to all the movie-makers. Keeping a movie accurate is something that could easily be done.
I am that guy that complains about everything tech in a movie :)
Like in an old French movie I saw a few days ago, the pilot of a fighter jet lost his engine and was almost crying and depressed until the squad leader came in front and showed the way to the airport, by flying in front of him. As if procedures were not already in place for this..
Cheers!
No doubt Star Fleet would have a fleet of ships in orbit around Earth. Hence the term fleet. This means that any ship in trouble would quickly get assistance and a tow into a proper old fashioned classical Newtonian orbit.
Interesting analysis you are sharing here, it clearly was a lot of effort, but I wouldn't mind too much if Start Trek makes some mistakes in its calculations, after all Star Trek is just a product for entertaining.
I wonder if you have seen Interstellar or some Black Mirror episodes?
Me too. I loved the movie and I wasn't trying to trash it. I just thought it was a good opportunity to provide education about physics and provide some real numbers.
No not yet. I will put them on my list of things to watch.