Drexxell's Falling Damage in D&D - Physics Based

(or "Have you ever wondered if using a d6 for every 10' fallen is realistic?")*

Last update 08/05/2016.

E-mail me your comments about this page

* For the impatient, the answer is "Yes". For those who want the details, see below.

Background: The table below reflects Earth-based physics, applied to falling damage in D&D. It makes the use of certain assumptions which, if altered, will alter this table and these results. These assumptions are: A fall is taken without any starting velocity and goes straight down, accelerating due to an Earth-type gravity (32.2 ft/s/s) Damage taken is directly proportional to the force of impact. The surface at the bottom of the fall is solid and non-yielding. A humanoid creature's bones and muscles "squish" downward approximately 1 inch (on average over the whole body) upon impact from any height. This squishing over 1 inch is how the force of impact is computed. A humanoid creature reaches a terminal velocity of 120 miles/hour (176 feet/second) if it falls long enough in an Earth-type atmosphere. At that point, air friction compensates the acceleration due to gravity, and no higher velocity can be achieved. The slow build-up of air friction as a function of speed is ignored until terminal velocity is actually achieved.

Mathematical Derivation A lot of the following is available in any Physics textbook or online. It is therefore presented here with only necessary detail. Because D&D uses 10 foot intervals for falling damage (1d6 per 10' fallen), the information below will stick to English units. Let "g" be the acceleration due to gravity. g=32.2 feet/second/second Let "y" be the distance fallen. It will be tabluated in 10 foot increments in the table below, and assumed to be known. Its relation to time fallen is: y=(1/2) × g × t × t (feet) Let "t" be the time it took to fall "y" feet. t=(2×y/g)(1/2) (seconds) Let "v" be the velocity reached after time "t". This is the velocity at the moment of impact. v=g×t=g×(2×y/g)(1/2) (feet/second) Let "dy" be the "squish distance" of the body as it hits. We set it to 1 inch: dy = (1/12) (feet) Let "vs" be the average velocity the body makes during the "squish". We assume that at the beginning of the impact, the body is traveling at "v" feet/second. At the end of the squish, it is traveling at 0 feet/second. As such, the value of "vs" is the average of v and 0, or: vs = v/2 = (g/2)×(2×y/g)(1/2)(feet/second) Let "dt" be the time it takes the body to squish the distance "dy". This can be computed by applying the average squish velocity, "vs", over the squish distance "dy": dt = dy/vs = (1/12) / [(g/2)×(2×y/g)(1/2)] (seconds) Let "dv" be the change in velocity from the beginning of the squish until its end. dv = v - 0 = g×(2×y/g)(1/2) (feet/second) Let "a" be the deceleration of the body during the squish. This is the change in velocity divided by the time it takes to squish: a = dv/dt = [g×(2×y/g)(1/2)] / {(1/12) / [(g/2)×(2×y/g)(1/2)]} = 12×g×y (feet/second/second). Let "m" be the mass of the falling body (slugs. Yes, "slugs"...hey, sue me for sticking to English units...) Let "f" be the force attributed to the deceleration of the body during the squish. This is equal to mass of the body times its deceleration: f = m×a = m×(12×g×y) (pounds) Let "k" be some real world to game world constant, relating hit points of damage to pounds of force. Its value will be unspecified. (HP/pounds) Let "x" be the damage taken due to the force of impact. This is equal to k times f: x = k×f = 12×m×k×g×y (HP) Note above that damage "x" is linearly proportional to distance fallen "y". That actually justifies the D&D system where 1d6 of damage is applied to every 10 feet fallen. To exemplify this, consider the following: Let "x10" be the damage taken by a body after falling 10'. x10 = 12×m×k×g×10 Let "x50" be the damage taken by a body after falling 50'. x50 = 12×m×k×g×50 The ratio of x50 to x10 = x50/x10 = 12×m×k×g×50 / (12×m×k×g×10) = 5 As in D&D, a 50' fall (5d6) is 5 times more damage than a 10' fall (1d6). Formula: For the purposes of the below chart, it is assumed that "12×m×k×g×10" pounds of force will be equivalent to "1d6" hit points of damage. (Thus were we able to 'hand wave' the lack of a definitive value of "k", as we can derive all falling damage by assigning the damage of a 10' fall to 1d6 and simply scaling everything up from there. Why Damage isn't Identical for Identical Distances Fallen: If the above assumptions are presumed to hold, then force on the falling body is identical for all falls of an identical distance. Why, then, should damage have a variable component? Why isn't a 10' fall always exactly, say 3 hit points of damage, every time? The use of dice to determine damage could be attributed to how a character lands. Although the force is the same, a character falling 10 feet and landing on their feet would likely take less damage than one who falls and lands on their head. Maximum Damage: It takes a fall of just over 480 feet to reach "terminal velocity" (176 feet/second). As such, in D&D, one can justify a maximum falling damage of 48d6 at 480'. Falls further than 480 feet will not yield more damage, as the body can not fall any faster. Mass-based damage: Astute readers will note that force (and thus damage) is also directly dependent upon the mass of the falling body, lending credence to the old saying "The bigger they are, the harder they fall". For DM's wishing to add a touch of "reality" to their games, a burly human (100 kg of mass) should take twice as much damage as a gnome (50 kg of mass). Two options leap to mind: Keep the Nd6 rolls (N = multiple of 10' fallen) and scale by race as: × 1 for Humans, Half-Elves and Half-Orcs × 3/4 for Elves and Dwarves × 1/2 for Halflings and Gnomes Change the die rolled based on race: d6 for Humans, Half-Elves and Half-Orcs d4 for Elves and Dwarves d3 for Halflings and Gnomes Example: Under these two different scenarios, a 50 foot fall for an Elf would yield: 5d6 x (3/4) = 4 to 23 (average 13) points of damage 5d4 = 5 to 20 (average 13) points of damage

Below, please find a table laid out for every 10' fallen, up until terminal velocity at 480 feet.

A second table is below this one, but laid out in seconds fallen.

Distance Fallen (feet)	Time it took to fall that far (seconds)	Velocity at impact (ft/sec)	Impact force with 1 inch deceleration (times 12×k×g)	Damage Roll
10	0.788	25.377	m*10	1d6
20	1.114	35.888	m*20	2d6
30	1.365	43.954	m*30	3d6
40	1.576	50.754	m*40	4d6
50	1.762	56.745	m*50	5d6
60	1.930	62.161	m*60	6d6
70	2.085	67.141	m*70	7d6
80	2.229	71.777	m*80	8d6
90	2.364	76.131	m*90	9d6
100	2.492	80.249	m*10	10d6
110	2.613	84.166	m*11	11d6
120	2.730	87.909	m*12	12d6
130	2.841	91.498	m*13	13d6
140	2.948	94.952	m*14	14d6
150	3.052	98.285	m*15	15d6
160	3.152	101.508	m*16	16d6
170	3.249	104.632	m*17	17d6
180	3.343	107.666	m*18	18d6
190	3.435	110.616	m*19	19d6
200	3.524	113.490	m*20	20d6
210	3.611	116.292	m*21	21d6
220	3.696	119.029	m*22	22d6
230	3.779	121.704	m*23	23d6
240	3.860	124.322	m*24	24d6
250	3.940	126.885	m*25	25d6
260	4.018	129.398	m*26	26d6
270	4.095	131.863	m*27	27d6
280	4.170	134.283	m*28	28d6
290	4.244	136.660	m*29	29d6
300	4.316	138.996	m*30	30d6
310	4.388	141.294	m*31	31d6
320	4.458	143.554	m*32	32d6
330	4.527	145.780	m*33	33d6
340	4.595	147.972	m*34	34d6
350	4.662	150.133	m*35	35d6
360	4.728	152.262	m*36	36d6
370	4.793	154.363	m*37	37d6
380	4.858	156.435	m*38	38d6
390	4.921	158.480	m*39	39d6
400	4.984	160.499	m*40	40d6
410	5.0463	162.493	m*41	41d6
420	5.107	164.462	m*42	42d6
430	5.167	166.409	m*43	43d6
440	5.227	168.333	m*44	44d6
450	5.286	170.235	m*45	45d6
460	5.345	172.116	m*46	46d6
470	5.403	173.977	m*47	47d6
480	5.460	175.818	m*48	48d6

Below is a table laid out with the same physics, but showing distance and velocity in 1 second increments, which can also be useful for in-game calculations.

* The distance fallen at second 6 is calculated assuming that the faller's velocity accelerates from from 160.8 feet/second to 176 feet/second in the first 0.47 seconds and then remains constant at 176 feet/second for the last 0.53 seconds. From that point forward, air friction counters the acceleration of gravity and the faller as at "terminal velocity", unable to fall any faster.