How would falling buildings affect the earth?

Question

So basically, humankind was at the pinnacle of technological greatness. We had a massive network of floating cities all around the globe. During "The Fall", all the massive spire-like buildings fell, impacting the earths crust with enough force to cause massive earthquakes. Assuming cities are made up of individual spires that are interconnected by skybridges and the like, is this plausible? Imagine huge nails being driven into the earth. Also, what would these cities look like 50 years later?

Answer 1

When an object at rest at some height is allowed to accelerate through gravity and hits a lower elevation, it converts its gravitational potential energy into kinetic energy and imparts the ground. There are considerations about elastic vs. inelastic collisions, but they don't matter much for this question. Things can be simplified down to the following formula:

$$E = mgh$$

Where:

• $E$ is the energy output
• $m$ is the mass of the object falling
• $g$ is gravity accelleration, which for Earth can be simplified to 9.8m/s²
• $h$ is the initial height

So just apply that to a city and see what gives.

The problem now is measuring the weight of a city. This in itself is an absurd question, and cities should have a range of masses encompassing various orders of magnitude. For an exercise, though, let's take a piece of NY for an example. Someone estimates the mass of Manhattan at around 250,416,934,000 pounds, which is 113,587,210,581 kilograms. Let's round it down to 10 $\times$ 10¹¹ kilograms, because this way we get a nice value in joules later on.

Drop Manhattan from ten kilometers up, and we get...

$$E = (10 \times 10^3m) \times (9.8 m/s^2) \times (10 \times 10^{11} kg) = ...$$

[redacted] that, I am rounding up accelleration to 10m/s² because I can.

$$E = (10 \times 10^3m) \times (10 m/s^2) \times (10 \times 10^{11} kg) = 10^{17} j$$

About 100,000,000,000,000,000 joules.

Just for fun, let's convert that to kWh. Why? Because that's the measure of energy used for TNT explosions, so you can lookup in Wikipedia for the TNT equivalent blast.

That amount of energy is about 27,777,777,778 kWh, so we are looking at a blast of about... 17% more output than the Fat Man, the atom bomb that exploded over Nagasaki by the end of WWII.

Well, not as spectacular a blast as I had thought for something of that size. But maybe that's because a city is made of a lot of hollow stuff - buildings need space for people, after all, so a city would act like a meteor of the same size but less dense. Remember, though, that's just the impact. Every flammable thing contained in the city will probably explode too, adding a little more boom to the whole effect. And the city will compress air below it as it falls like a meteor, so there will also be a lot of fire.

In the end you are left with a lot of red-hot rubble. Few structures should be recognizable, if any pieces are left intact at all. All organic material will evaporate and, if not confined in a closed space, reach the atmosphere to later rain in the form of black droplets with the consistence of petrol. The ground around the blast site will probably become vitrified.

Fifty years later, you have the same rubble cooled down to the same temperature as the surrounding environment.

P.s.: of course, you can always double the drop height to double the energy output.