Two Giants Making Love

The Problem

So I'm a fan of solving ridiculous physics problems, it's always been a hobby of mine. A friend of mine recently posed a rather lewd question. If two giants were making love on earth, how tall would they need to be to crack the earths surface open and sink into the mantle. Well. Let's get started!

The Assumptions


Whenever solving any physics problem we must start off by noting our assumptions. My assumptions are as follows:
  1. Height is proportional to volume so increasing height also increases width and breadth with the same proportion.
  2. The giants have unbreakable bones, infinitely strong muscles and are immortal.
  3. The crust is of constant density. (We'll use the average density given by Wikipedia of 2.83 g/cm3 or 2830 kg/m3)
  4. The earths shape is irrelevant for this question.

The Solution


So in order to answer this question we need to know two things. First how much energy is required to get through the crust of the earth to the mantle and second how much energy giants of a certain height generate. We can then combine the two equations to get a solution. First let us ask how much energy is required to break through the earths crust.

Cracking the earth


I admit it, I'm not entirely sure the best approach to solve this problem. I've seen a few approaches online (such as the the amount of energy required to move two earth hemispheres apart) but these approaches don't seem to address our current problem. I decided that we're dealing with scales such that the chemical potential energy required to crack rock is relatively small and negligible. So instead the energy required to crack the earth would be roughly equivalent to the energy required to displace the crust somewhere else. Basically, how much energy will it require to dig a hole? 

First of all lets ask a question, how much energy would be required to move a kg of crust from the mantle to the surface. This is equivalent to the difference of gravitational energy between the bottom of the crust to the top. The crust (according to Wikipedia) is between 5 to 50 km thick. I'm gonna use 50 km as the thickness for our absolute worst case scenario. In that case the difference in potential energy is given by the (negative of the) gravitational potential energy:


Where the values are as follow:
  • G: Universal Gravitation constant or 6.67 * 10-11.
  • Me: is the mass of the earth or 5.97 * 1024 kg.
  • m: The mass of our object or in our case 1 kg.
  • r1: The radius of the earth or 6.378 * 106 m.
  • r2: The radius of the mantle, which is r1 - 50km or 6.328 * 106 m.
Plugging all these values in we get our required gravitational energy to be 468,814 J. That's about the same amount of energy as a large automobile moving at highway speeds, so not so much! However, we're trying to displace more than one kilogram, we need a hole wide enough for both our giants to fit in, so how big do we need to fill in this whole? It turns out this value is directly related to the giants height, which is what our question is asking.

First of all a humans arm span is roughly related to the humans height (according to Wikipedia), we'll need a hole large enough that our giants can't get out even if they outstretch their arms so we'll use this as the minimum length of the hole. Calculating the average width of the hole actually turns out to also be the giants height, because as the giant sinks into the hole they'll be lying flat. So for a single giant the hole's minimum dimensions is simply r2 (where r is the giant's height) but because we have two giants we need to make a little bit more room. So to be on the safe side and assume these two giants aren't necessarily cuddling the entire time they're sinking we're gonna give them plenty of room and double our minimum length and width. Thus the minimum area of this hole needs to be 4r2.

Ok, so now that we know the minimum area we need to calculate how much mass we'll need to displace. We'll assume for simplicity sake that our hole is literally just a big square hole in the earth that goes straight into the mantle. We can calculate the total mass needed by simply thinking of the volume (area multiplied by height of the crust) multiplied by the average density of the crust given in our assumptions. This gives us our equation:


Plugging in the numbers we get a final result of m = 4r2 * 1.415 * 108 kg. That's a lot of mass! To figure out how much energy we need to displace the maths we notice our potential gravitation equation from earlier and we see that the energy required is proportional to the mass. So all we need to do is multiply our new mass energy required to lift 1 kg we derived earlier. This gives us

E = 4r2 * 6.634 * 1013 J

We will call the constant k (k =  6.634 * 1013 J) for simplicity reasons. Thus our final equation for the energy required becomes:

That's pretty simple isn't it!

Making Love!


Now for the fun stuff, we need to know how much force our giants are making when they make sweet, sweet love. To calculate this we need to know how much force an average human makes when they make love, then we need to scale that force up. For the purposes of this video we will be assuming the giants are directing there... thrusts... directly towards the earth and so all the energy of the thrust will be directed into the crust.

Ok to calculate this we need to know a few things. First we need to know the average mass of a human and more importantly how that mass changes with height, then we need to calculate the velocity of a human thrust and that's all we need to calculate the total kinetic energy.

The mass is easy the average mass of a human is 62 kg according to this paper. We also know that mass is simply volume multiplied by density and if we assume that the density of our giants doesn't change (don't forget, unbreakable bones and all that) we just have to scale our mass the same way we scale our volume (remember, we've decided to scale all dimensions of our giant equally). Because the average height of a human (according to this website) is 178 cm (for a man) we can say the mass of a 1 m tall human is 34.83 kg (we'll call this value maverage). This means the mass of our human is given by

Now we need to figure out the velocity of the... thrust. I will be honest this calculation is gonna be a little... rough (hehe) because getting accurate data on the velocity of human sex is not readily available. So we need to make some assumptions and basically try to get a gauge of how much distance is travelled during a single thrust and how long a thrust usually takes. Getting the distance is relatively simple we just need the average length of... well... a penis which turns out to be (according to this website) 5.6 inches or 0.142 metres.

Now that we got that out of the way how long does a thrust usually take? Well according to this website a human generally thrusts every 0.8 seconds during sex. Now we're gonna take the guess that the time to thrust is insertion then resetting back to original position. While it's probably true that inserting and resetting don't take the same amount of time, there's no way to verify the exact ratio with information I have on hand so we'll play it safe and just say the insertion is half the overall time.

That means it takes 0.4 seconds to travel 0.142 metres during a sexual thrust indicating a velocity of  0.355 m/s. Now remember in my assumptions I said that everything scales, EVERYTHING scales, and because the giants muscles are infinitely strong we're gonna assume for simplicity sake that the time it takes for a giant to thrust doesn't change, but the distance required to thrust does. Normalising again for a 1 metre tall human we get a velocity of 0.199 m/s (which we'll call vhuman) which means the overall velocity of the giant is:


So the total kinetic energy of two giant's having sex is given by the kinetic energy equation of half mass time velocity squared. There are two giants so we double the mass just giving us mass times velocity squared, which gives us the final equation of:


In this equation we're ignoring the force applied to the ground by the sheer weight of the giants because that force is going to be proportional to r3 rather than r5, so it'll probably be small. This is more fun anyway because it only considers the sexual thrust.

Putting it all together


So we now have our equations for the energy our giants will produce and the energy our giants will need. To get the minimum energy required to make their hole we just make these equations equal to each other and we get:


Solving for r we get:


Plugging in those numbers and we get the final answer.

The Result

A giant needs to be 615,688 m tall in order to crack the earth by making love.


...And now we know.




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