Deep Impact is a 1998 disaster film, one of two Hollywood blockbuster disaster films released that year (the other was Armageddon starring Bruce Willis). The plot revolves around a teenage astronomer (a young Elijah Wood) who discovers a comet on a collision course with the earth, and the earth’s efforts to avert the disaster. Disaster films always contain lots of mathematically-based scenarios, which we’ll get into in this tutorial.
The famous scene where the comet hits the earth can be seen here:
Math Scenario 1 – Detecting the Comet
In the film an amateur teenage astronomer detects the comet. The comet itself is 11 km across, which is quite small in terms of objects that we normally look at in the night sky.
The formula for the angular resolution of a telescope in radians is:
Resolution = wavelength / telescope diameter
An average visible wavelength is 550 nm (nanometres), or 550 * 10e-9.
We can work out how far away an amateur telescope with a diameter of 300 mm would be able to theoretically see the comet:
Resolution = 550e-9 / 0.3
Resolution = 1.833e-6
We need to find out at what distance an 11 km wide object can just be detected using this angular resolution. For very small angles like this, we can use the approximation that tan of a very small angle given in radians is just the angle itself e.g. tan(a) ~= a for very small angles of a given in radians.
tan(resolution) = opposite / adjacent
tan(1.833e-6) = 11 / d
d = 11 / tan(1.833e-6)
d = 11 / 1.833e-6
d = 6001091 km
So the amateur astronomer might have got lucky and detected the satellite about 6 million kilometres away. That’s not very far away at all – if the comet was on a direct collision course with earth it might hit it very soon, let’s say it’s moving at 40 km/s:
Time to impact = distance / speed
Time to impact = 6001091 km / 40 km/s
Time to impact = 150027 s
Time to impact = 41.67 hrs
Now in the movie, the time to impact is more than a year. However, there’s a possible explanation for this – the astronomer could have detected the comet as it passed by the earth, with the impact to happen at some future time after the comet had gone around the sun.
Math Scenario 2 – Nuking the Comet
The maximum range of Intercontinental Ballistic Missiles is around 10000 km – but that’s for ballistic flights between locations on the earth. For curiosity’s sake, let’s say the missiles could be launched 10000 km into space to intercept the comet. That’s only a tiny distance compared to the 400,000 km to the moon.
If the comet is incoming at a speed of 40 km/s, how far from the earth should it be when the missiles are launched to maximize the interception range? Assume the ICBMs have an average speed over the journey of 4000 km/hr.
First of all, the missiles will take some time to travel the 10000 km – so we should launch before the comet reaches that range from the earth.
We can calculate how long the missiles will take to reach the 10000 km mark:
Time to reach maximum distance = distance / speed
Time to reach maximum distance = 10000 / 4000
Time to reach maximum distance = 2.5 hrs
Now we can work out how far the comet will travel during that time. Don’t forget to use consistent units:
Comet travel time during missile travel time = comet speed * missile travel time
Comet travel time during missile travel time = 40 km/s * 2.5 hrs
Comet travel time during missile travel time = 40 km/s * 2.5 hrs * 3600 s / hr
Comet travel time during missile travel time = 360000 km
So we’ll need to launch the missiles when the comet is 360000 + 10000 = 370000 km from the earth.
Math Scenario 3 – Outrunning the Tsunami
When the comet hits in the Atlantic Ocean, it generates a massive tsunami measuring about 1 km high that devastate the coasts in Europe and the American continents.
The main character Leo along with friend Sarah and her baby sister are on a trail bike running for the hills (literally!).
The bike has a top speed climbing uphill on the rough trails of 15 km/hr on a slope gradient of 1:5. If they have 10 minutes before the wave reaches their area, we can calculate how much altitude they can gain.
Altitude gain = rate of altitude gain * time
The rate of altitude gain depends on the bike’s speed and the slope ratio. We can calculate the angle a of the triangle shown in the diagram here, and then use it to work out the vertical speed component:
tan(a) = opposite / adjacent
tan(a) = 1 / 5
a = 11.31 degrees
Now work out the vertical speed component:
sin(a) = vertical speed component / bike speed
vertical speed component = bike speed * sin(a)
vertical speed component = 2.942 km/hr
Now multiply this by the time duration to get the total vertical climb:
Altitude gain = 2.942 km/hr * 10/60 hrs
Altitude gain = 0.4903 km or 490.3 m
Real Life Example – Extinction Level Events (E.L.E.)
Extinction Level Events are events that lead to a major reduction in living things on our planet. Such events are rare, occurring maybe a dozen times over the past 500 million years.
One of the most famous extinction level events was the one that led to the extinction of the dinosaurs about 66 million years ago, at the end of the Cretaceous period. About three-quarters of animal and plant species disappeared rapidly.
What led to the event is still not certain. The most supported theory is that a massive comet or asteroid hit the earth causing a global protracted winter that severely affected the ability of plants to perform photosynthesis.
If the chance of an extinction level event occuring in any one year is 1 in 50 million, we can calculate the chances of you not having to deal with one during your lifetime. Let’s say you live to 120 and are already 15 years old:
Chance of no ELE during your life = (chance of being safe per year)^(number of years)
Chance of no ELE during your life = (1 – 1 / 50000000)^(120 – 15)
Chance of no ELE during your life = 9.999979
That gives a 0.00021% chance of there being a major extinction level event during the rest of your lifetime – hopefully those odds are realistic!