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Armageddon: The End of Physics (and life) as we Know it

Armageddon is one of those movies that everyone knows about I think? There's no doubt that it's at least a half decent movie and one worthy of watching if you've got a spare 2 hours. However, the whole "let's blow up an asteroid with a nuke and have the two halves of the asteroid miss us entirely" thing, is utterly false and impossible to do. In actual practice, one nuke would barely separate each of the asteroid pieces a football field apart from one another by the time it reaches the movies "0 barrier".

However, it does pose an interesting question. In the event that earth does come into the crosshairs of an "extinction level" asteroid, what on earth (no pun intended) do we do?

The Plan
So, although throwing nukes at the thing last minute would give a great last-minute fireworks show, it won't work. If we are going to actually hit this thing with nukes, we need to be able to do it way in advance. I'm talking 10+ years in advance. We also need to know it's going to hit us probably 20+ years in advance so that we can have time to build and launch the equipment needed to divert the asteroid. The leftover 10 or so years would just be travel time for the asteroid to get farther and farther off course. And, just to eliminate the obvious problem we all should have with trying to launch a nuclear weapon into space (god knows what could go wrong during launch there), we could just simply hit it really hard with another object. It's like a really really big game of pool, where the cue ball strikes the 8 ball and you win $20 from your friend and you save the earth. 

Basically, we are going to hit off course by propelling something of great speed into it. 

The Math behind the Plan
We can be assured that this plan will work through the conservation of energy. Basically speaking, it states that the final mechanical energy of one system has to be equal to the initial mechanical energy of the same system. And, the mechanical energy is expressed as the sum of the Kinetic Energy and Potential Energy. This can be written as such. 

MEf = MEi
Kf+PEf = KEi + PEi

Using this, and some basic kinematics, we can find out how much energy we would have to apply to the asteroid to get it to miss us. 

Raw Data and Calculations
First, assuming that this asteroid is heading straight for us and only traveling in the X  direction as a result, we need to find how much we have to make the velocity in the Y direction be (Vy) in order for the asteroid to miss us. We can do that using the following equation and info. 

Δy = Vyt
Δy/t = Vy
Where Δy = 384,000km (distance from earth to moon)
t = 10 years = 3.2 x 10^8
So, Vy = 1.2m/s

Next, we need the mass of the asteroid. We are going to try to solve for 2 different asteroids here. The first one is going to be the asteroid from the movie Armageddon (dubbed "dottie"). Meanwhile, the second one will be of similiar composition of materials but it will be 100km in diameter (the minimum needed for an extinction level asteroid is estimated to be 95km). We will denote this new asteroid with the name "jack" because jack sounds like a good name for an asteroid. 

Ma(dottie) = Density * Volume
Density = 5500kg/m3
Volume = 4/3πr^3 = 1.0 x 10^18 m^3
So,
Ma(dottie) = 5.5 x 10^21 kg

Similiarily, 
Ma(Jack) = Density * Volume
Ma(Jack) = 5.775 x 10^13 kg


Now, keeping both of those things in mind, let's move on to the actual Conservation of Energy part. Using this we are going to solve for how much energy we need to pack into our projectile to hit the asteroid off course. 

Assuming Vy(asteroid) = 0m/s
And, Vx(projectile) = 0m/s

1/2MaVx^2 + 1/2EP = 1/2MaVxf^2 + 1/2MaVyf^2
=>
EP = MaVyf^2

EP = Energy of projectile
Ma = Mass of asteroid 
Vyf = Final Y Velocity needed of the asteroid

So, to start, let's plug in the value of mass for our new asteroid, Jack.

EP = (5.775 * 10^13kg)(1.2^2)
EP = 8.316 x 10^13 J

This kind of power can be achieved by hurdling a
rocket the weight of the Saturn V rocket (50,000kg)
at about 41,000m/s.

So, after a little bit of math, we can see that it would be possible to hit the object with enough force. However, from a technical standpoint, it would be extremely hard to create a craft that was that heavy and could fly at that speed. 

However, let's say that the asteroid from Armageddon, Dottie, was going to hit us. How fast would we need to hit it with the same craft?

EP = (5.5 x 10^21kg)(1.2^2)
EP = 7.92 x 10^21 J

This translates to a necessary speed of about 380,000km/s
assuming that the object had a mass of 50,000kg. 
This is actually impossible because it is faster
than the speed of light = 300,000km/s

So, even if we hit the asteroid at the speed of light, it wouldn't be enough. To actually save ourselves from an asteroid of this size. We would need to be able to hit 30+ to 40 years in advance. 





Comments

  1. Wow! Excellent! This is exactly what I was looking for in this assignment.

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