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I have a character who has managed to get herself a movement speed of 100'. She is a centaur, and as such specializes in lance based combat and charges. Now in the real world a lance moving faster would have more power behind it. With this centaur moving at 56 MPH and putting her 1473 pounds behind that lance it is striking with a force of 16,726.39 Newtons concentrated down onto the lance tip, which is something rather tiny like a hundred thousandth of an inch. This results in the lance point having 95,510,148 PSI when it strikes a target. This is enough force to pierce up to 8 inches of modern ultra-high carbon steels (assuming no loss of power by friction). In other words that lance is going into and through just about anything that gets in its path.

Now is this knowledge just a fun bit of coolness to know or does D&D reflect physics and give a damage boost for speed of a handheld piercing weapon?

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No, movement speed has absolutely no relation to attack or damage.

Considering that there are ways to boost speed to stratospheric levels (literally, in the sense that you can exceed Earth’s escape velocity; before an erratum, you could even exceed c), that’s probably for the best.

Instead, 3.5 and Pathfinder model the extra oomph you can put behind your attacks when moving at high speed via the Charging combat maneuver. Charging gives you a +2 to attack (and a −2 to AC), but you get the same benefit when you charge no matter if your speed is 15 ft. or 150 ft. And you don’t get the benefit without charging, even if you move farther in a single move action than someone else might while charging.

The only innate effect of high movement speed, aside from the ability to move further in a round, is the (rather large) bonus you receive to Jump checks.

Specific feats and class features may reference movement, however. For instance, the Scout (Complete Adventurer) has a Skirmish class feature, which deals extra damage as long as you move 10 feet, and can take the Improved Skirmish feat for additional bonus damage if you move 20 feet. The Expeditious Dodge feat (Races of the Wild) gives you an AC bonus as long as you move 40 feet.

I haven’t found any options yet that scale with speed, but as I pointed out, speed also increases Jump checks. The Roof-Jumper feat gives you +1d6 damage per 10 feet you fell from above the target. If you have a sky-high Jump check, then you can jump, well, into the sky – and stack extra d6’s on your attack. Note that you’re not allowed to slow your fall (e.g. feather fall) if you want the damage, though, which means you may take falling damage (Tumble, as well as certain items, can mitigate falling damage, however). All that said, however, each 10 feet of vertical jump height requires a DC 40 higher, and each +40 bonus to Jump from movement requires 100 ft. more movement speed. You can get d6’s of damage much more easily than you can get 100s of feet of movement speed...

See Are there rules for dropping on an enemy as an attack? for more ways to get extra damage from jumping (though only Roof-Jumper directly scales with height).

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Weren't there some maneuvers in 9 swords that scaled with jump? – starwed Aug 21 '13 at 23:32
Scaled with? I don’t think so. I’ll double-check; several require jump checks for extra damage, but they don’t scale if you beat the DC by extra, which makes most of them a little removed from this question. Certainly valid in the linked question about jump-attacks. – KRyan Aug 22 '13 at 0:37
This question's very like the old commoner congaline railgun one: It's all predicated on the idea that D&D attempts to model physics in some vaguely realistic fashion. – GMJoe Nov 26 '15 at 2:59

The feat Spirited Charge lets you deal double damage on a charge with a melee weapon, or triple damage with a lance. However, this only applies while mounted, which technically a centaur is not.

Rhino hide barding would give your centaur 2d6 bonus damage on a charge.

Sadly, speed does not translate directly into damage. Otherwise, the peasant railgun would be the ultimate weapon: line up 2000 peasants, have them pass a spear from one end of the line to the other, and when the spear gets to the end it's moved 10,000 feet in 6 seconds, thus exceeding the speed of sound.

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Haha, peasant rail gun, I love it =D – Styphon Aug 21 '13 at 20:04
Did you read about the railgun in 'Harry Potter & the Natural 20'? I think I saw it there. – Dakeyras Aug 21 '13 at 21:32
While being a centaur is not being mounted, I would have to accept a centaur as a "virtually mounted" character. They have the movement and the height that mounted characters would have. Also, they have better control of the "mount" in this instance as well. Just my two cents. – CrimRei Aug 21 '13 at 23:08
I remember thinking on the berserker expresso on the kobold monorail. Lineup thousands of kobods, with supreme cleave and the base damage a frenzied berserker does, and since you just move 5' with each kill... All aboard! bloody berserker train is about to depart! – apacay Jul 4 '14 at 2:51

No. D&D and Pathfinder do not come close to simulating real physics.

The only place, I know of, where you get variable damage based on movement speed/distance is fall damage.

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Unfortunately not, neither D&D 3.5 nor pathfinder simulate any kind of damage based on speed, except for possibly falling damage. To simplify things anyone charging with a lance does double damage automatically on a successful hit.

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Sadly, no, Pathfinder does not give you extra damage for base speed. However, you do get increased range for charge attacks. You see, there are a group of actions a creature can take a round called Special attacks and Combat Maneuvers, that any character can take for either a standard (an attack) or full-round action (your full turn). Among these inherent abilities is the humble charge, which allows you to move at double your base speed (in a straight, unhindered line) and have an increased chance to hit your opponent. Furthermore, since your centaur has a lance, a landed charge deals double damage!

On a semi-related note, you could have racial access to mounted feats that improve your charge attacks (since everything below your waist is a horse). These include abilities that let you make a spring-attack during your charge, deal extra damage and let you make a large turn while charging.

TLDR: Speed does not improve damage. Speed improves Charging distance. Feats and items improve charge versatility and damage. Talk to your DM to see if you can purchase Mounted Feats to further advance your character.

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While I can not think of any extra damage other then skirmish, you could always combine scout and the dervish. Then if you take combat reflexes and karmic attack you will have a lot of extra attacks as you move into and out of other people threatened square, virtually giving you more damage based on your movements. Also, robilars gambit will also net you some more AOO if you so desire. Some people will try to tell you that you cannot use Karmic Strike and Robilar's Gambit together. That is not true as they have different triggers.

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RAW/RAI? No, nothing based on speed.

Should you house rule it? No. 56mph is not that fast. Thoroughbreds (which I think would be pretty similar to a warhorse) run at around 40 mph and racehorses run at 50 mph.)

A barded warhorse with an armoured rider will way pretty much identical to a centaur.

Lance rules are balanced around you weighing a lot and travelling at 40 mph, that is why you do double damage on the charge, in addition to having the +2 to hit (which can become +3 damage if you use power attack).

If your person gets an actual fast movement speed (mounted on a dragon flying at 250 fly speed) then there might be an argument that it is realistic to house rule it. However, I would stress that DnD weighs much more heavily on the fun and (sort of) balanced side of things, over realism.

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Besides flying there is no speed based damage increases due to force of impact. D&D is just not that realistic outside of the individual DM's mind.

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I'm going to downvote this answer, since it doesn't include any proof - don't take it personally, and if you include some evidence that shows this is correct I'll happily reverse my downvote. – Miniman Nov 26 '15 at 23:35

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