D&D 5e: The Falling Flyer Problem

From reddit:

In a session recently my players asked for a ruling on fall damage when it came to flying creatures. RAW fall damage would be equal to the number of feet a target was from the ground no matter how fast they were travelling prior to that. The issue came up where I was attacking with a dragon that had a fly speed of 80 and one of my characters used his turn to paralyze it using a houseruled effect that is basically a copy/paste of hold monster. Since the dragon was paralyzed it couldn't continue flying and fell. It was strafing them at 30ft in the air so RAW it would have taken 3d6 damage, but he thought that was unfair given that it had moved (80ft) and then dashed (an additional 80ft) showing that it obviously was moving at 160ft per round, which if converted to fall damage would be 16d6.

TlDR: Is fall damage RAI velocity and therefore equal to ft/round.

There were some very misinformed responses to this question.  The bottom line is that the horizontal velocity component given here as 160 ft per round does impact the physics of the problem to increase the effective falling damage.

I'm not sure I have mentioned it elsewhere, but I am a physicist, which comes in very handy as a GM.  Being a physicist, I just cringe when I see some of the tomfoolery pulled with physics equations on place like reddit.  I spent a good bit of time the other day trying to debunk the top post for this question the other day.  Finally I gave up and decided to put together a full analysis here, that I'll probably link over.  Don't let the word physics bother you, I'll speak plain.

Please reference this article for a better explanation of the full equations.

First let's revisit the RAW: for falling damage we take 1d6 bludgeoning damage per 10 feet of falling given on page 183 of the D&D 5e Player's Handbook.  Maximum damage is  at 20d6, which we will assume indicates some general "terminal velocity" limit. 

When objects fall, gravity causes them to accelerate down.  Accelerating down doesn't change their horizontal velocity, but it causes them to fall faster and faster.  This would continue without limit except that the drag of air opposes the force of gravity, meaning an object will only get faster and faster until the drag force equals the force due to gravity.  This velocity is called the terminal velocity.

Falling doesn't really do damage.  The problem is that when an object with velocity hits the ground, it gets acted on by a force from the ground that causes it to change velocity to zero.  This force causes damage.  The faster an object is falling, the more damage it takes.

Now it doesn't really matter too much which direction this velocity is in so long as the ground causes it to stop quickly.  So our flying dragon may hit the ground at an angle, but we don't really care, because he is going to sink quickly into the ground and stop.   

Now let's do a little math.  The acceleration due to gravity is 32.174 feet/second/second, meaning that our velocity increases 32.2 feet per second for each second we are falling.  Using this, we can determine the speed at which we hit the ground when falling for various amounts of time.  We can numerically integrate our velocity to calculate the height that we've fallen at each time, as well.  Based on RAW, we can then calculate our damage.  The image below shows these relationships.

Here we can see that in about 3.5 seconds we fall 200 feet resulting in our maximum damage.

The figure also illustrates another big problem though.  The damage should be proportional to the velocity when we hit the ground, but instead it is proportional  to the distance we fall.  This is incorrect per physics, but is a nice simplification for the game.  To include damage from horizontal velocity, we need to know how much damage per unit velocity we are using.  We will calculate this at 10d6 damage value, the midpoint, to get 0.124 d6 per feet per second. (We could have used the point at other values or an average, but this value is reasonable.)  This midpoint linearized model now gives us the ability to convert velocities to damage and back again using a similar approximation as the RAW.

How fast is our dragon moving in feet per second?  160 feet in 6 seconds (1 round) which is 26.7 feet per second.  This means the damage from horizontal velocity alone if we crashed would be more than 3d6.

Now, how do we combine the effect from horizontal together?  It turns out to be a vector addition as given below.

The Vector Sum of Velocity Vectors

To get the magnitude of the vector, which is our crash speed, we take the root sum square of the two values.  So in our case, we had 26.7 ft / sec flight speed horizontal with a falling velocity of 43 ft / second.  The total speed is now 50.6 feet per second.  Multiplying this by .124 d6 of damage per feet per second gives us damage of 6d6, twice what we predicted from falling alone.

Now as a GM, how do we really want to rule this?  After all, we really don't want to build a full kinematic model for every case.  The "average" rule would be add a d6 of damage for every 50 feet of motion of an object moved in the round before it started falling.  A normal character walking off a cliff takes no extra damage, but a normal character dashing off a cliff could likely take an extra d6 of damage.