After many days, it was determined that what was really needed was the terminal velocity of a balrog.

More belrog debate here and here

Seeing that we need the mass of the balrog before we can continue, our brave author buys the only scaled balrog model she can find. The theory being that if she can measure the volume of water the model displaces to calculate the mass of a full size balrog.

This bring ups some interesting thoughts…

• How tall is a full size balrog?

Toosdoodle says: In the Two Towers Merry and Pippin discuss their exact height when Pippin drinks the special water in Treebeards forest and it makes him get taller. You could determine how tall Gandalf is based on Pippins height and then Compare Gandalf to the balrog to determine the Balrogs height.

• [@more@] Do you have to build the model before you measure the volume of water displaced?

Anonymous says: Do you really need to build the balrog? Surely if you immerse all his dis-embodied parts it will be equal to the volume of all his parts glued together.

Sparkzy says: You know, you don't need to assemble the Balrog. Just make sure the hollow pieces are filled in with something waterproof (spackle?), and then dump the pieces in the water. It would be the same as if you chopped up a Balrog, then determined its volume. Should be the same as an intact Balrog.

• Is a Balrog's terminal velocity slower then a falling Ainu?

Fings says: A rough approximation: Gandalf falls after the Balrog, but catches up with him, so obviously the Balrog's terminal velocity is slower than a falling Ainu (which for purposes of estimation, we will assume equals a falling human). As the terminal velocity of a falling human is around 50m/s on the low end, let us assume that speed for the Balrog.

Furthermore, let us ignore the initial acceleration and assume the Balrog starts at that terminal velocity. Therefore, 50m/s*104s = 5200m, or 5.2km. This is a rough upper bound, as the Balrog may have an even lower terminal velocity, or had its speed reduced from bouncing off walls, etc, plus the fact I ignored the slower than terminal velocity for the first few seconds. So I would say the lake is no more than 5.2km down.

• Then again, maybe we aren't taking in all the variables?

Steve V. says: To add more complication to all this, let's go back and forget about terminal velocity and realize that there are other things happening if you can accelerate towards the center of the earth for 104 seconds.

In this case gravity is not constant, but is decreasing towards the center. Offhand, that makes a difference of about 2% in the gravitational force at the end of the plunge – mostly negligible. What would happen if it went to the center? Weightlessness. And if the hole goes all the way through? Periodic motion down to the center, up to the other side to zero speed, then reverse and start back all the way through. How many seconds to get to the center? Need to do an integral…

Then another thing is forgotten. The radial velocity – the earth is spinning at the same angular velocity, but the linear velocity varies based on radius. The balrog's sideways motion does not vary with radius (depth) and the vector direction was determined when he started falling, but the hole orientation is rotating with the earth. The balrog would presumably hit the wall when falling far enough. Need to do the complete math here – the real thing is that the balrog is doing a truncated orbit… Have fun.