Newton's Laws of Motion

1. An object at rest tends to stay at rest. An object in motion tends to stay in motion along a straight line unless acted on by an outside force.
2. Force equals mass times acceleration.
3. Every action has an equal and opposite reaction.

Take a careful look of the definition of force. If there's no acceleration (that is, no change in speed), there's no force. A person falling at terminal velocity has no forces acting on him. When he hits the ground and suddenly decelerates to zero velocity, though, there's a whole lot of force acting on him. That's why landing hurts.

Spells that move things - teleports, flight, levitation, telekinesis - come into the realm of Newtonian motion. The usual arguments start over Newtonian reference frames, which we'll get to in a bit. Equations of motion derived from Newton come up in discussions of falling.

Newton's First Law is why a mage had best be careful when firing off a fireball. Once that little fireseed is launched, it moves in a straight line towards the mage's target. If something - say an ally's shield - should suddenly get in the way, the mage can't bend the fireseed's path to arc around it, unless he could telekinetically shove it away from the obstacle. It will tend to move in a straight line until acted on by a force. Law Three ensures that hitting something generates a force back, so the fireseed explodes into a fireball on impact.

Law Three is not the same as the Law of Conservation of Momentum, although they seem similar. Newton's Third Law involves forces; conservation of momentum involves, well, momentum. The Third Law says that if I apply the force of my weight (my mass accelerated by gravity) to the floor, the floor responds with an equal force. If the floor couldn't, I'd fall through it.

So what happens when a levitating or flying mage passes over a trapped floor? In Robert Asprin's "Myth" books, the flying wizard actually "pushes off" of the ground with his mind. This push can set off pressure-sensitive traps! D&D mages generally don't; maybe they're "pulling up" on the ceiling? Well, we know they're not doing either; it's *magic,* after all. But if you want a scientific explanation for the phenomenon, they've got to either push or pull on something to lift themselves off the floor.

The Second Law (F = ma) doesn't show up directly very often. Mostly, people like to confuse forces with energy or momentum and then argue about it. A force is a push or a pull. It can be a sudden, sharp push or pull - that's called an impulse. It can also be gradual, like a car being "pushed" by its motor and speeding up. Energy is the ability to do work, and work is applying a force over a distance. So a rock on top of a cliff has stored (potential) energy to push on me very hard if it were to fall. But the rock can have lots and lots of energy just sitting there, without necessarily exerting any forces. And momentum is mass times velocity - a measure that would be pretty useless if it weren't conserved in a system (which is a pretty nifty thing).

Falling

How many PCs have, intentionally or not, found themselves plummeting towards the earth at 9.8 m/s^2? And how many of them have turned to ask the GM: "Can I cast feather fall/activate my ring/create a makeshift parachute from my cape?"

Here, the GM has a recourse to physics. Say the PC is a height h above the ground. Say also that he begins his fall with some vertical velocity, v0. For simplicity, let's also say that v0 = 0 and the PC isn't trying to leap up or enter into a power dive. If you're using English units and a standard Earth-like gravity field, you can estimate gravity as 32 ft/s^2 or even 30 ft/s^2 if you don't mind rounding; those with the good sense to use metric can use 10 m/s^2. Let time (in seconds) be t.

Here is the equation you will want to use:

h = 0.5 * g * t^2

g is the value you picked for gravity, and t^2 means "t squared."

Assuming your PC is just running straight off the top of the cliff, then he has [square root(2*h/g)] seconds to do whatever it is he wants to do. For a 100-foot tall cliff, this is about 2.5 seconds. While that might be enough time to activate a magic item or innate ability, the PC will be hard-pressed to cast a spell or perform any action before he meets up with terra firma.

Traveling

More mundanely, the venerable old "travel at constant velocity" equation is fine for determining how far your PCs go per day:

distance = velocity * time

Make sure your units match. If the PCs walk at 3 miles/hour for 20 minutes, they do not travel:

d = 3 * 20 = 60 miles

You must either convert miles/hour to miles/minute or minutes to hours. In this case, it's easy to see that 20 minutes is 1/3 of an hour. So the PCs actually go:

d = 3 * 1/3 = 1 mile

Much more reasonable, isn't it?

Teleporting and Frames of Reference

Some GMs like to rule that a mage cannot teleport into a moving vehicle, like a carriage or ship. The logic is that the moving object is not where the mage last knew it to be. Physics-saavy players will retort that the whole universe is expanding anyway, and the planet is circling the sun, so everything's in motion, and teleport still works. So they should be able to 'port where they like, right?

Maybe.

If a body is moving at a constant velocity, we can call it a "Newtonian reference frame." This means that, if we were inside such a body with no way of looking outside, we couldn't determine if we were moving or standing still. Try this for yourself. Sit very still at your computer. Does it really feel like you're moving through space at hundreds of miles an hour? Of course not. You're in a Newtonian reference frame - in this case, your planet.

A GM who wants to make 'porting into moving objects difficult yet allow 'porting on a moving planet might consider the following option: allow a mage to teleport within a Newtonian reference frame without problem. So a mage on a moving ship can teleport to all locations within the ship as easily as a mage standing on a mountaintop can teleport to his keep half a world away. Let's look at what happens when a mage tries to move between reference frames:

A mage on shore tries to teleport to a moving ship. Unless the GM allows her to "lead" the ship with her spell in an attempt to compensate for the movement, she'll teleport to a spot where the ship was when her spell went off. If the ship is slow and large, she may still end up on it, just not where she thought she'd be. Otherwise, she may be rather wet and unhappy.

The damp mage decides the nautical life is not for her and wants to teleport from the moving ship back to shore. The GM may rule that the ship imparts its velocity to her so that she hits the ground running - or at least propelled in the direction of the ship's travel. (Imagine jumping out of a moving car and trying to hit the ground running). The GM may want to make a rule for "horizontal falling damage."

What about an accelerating mage? Say our unfortunate example finds herself falling off a cliff - a really big, big cliff, so she'll have enough time to cast. She's in the "world frame," since she's surrounded by cliffs and other planetary features. But she's not stationary with regards to it. By the logic described above, she can't teleport. Just can't - the spell won't work unless the mage is standing still in her frame of reference. A nice GM might allow her to teleport nearly safely to the ground, maybe just inflicting falling damage for the distance she'd fallen before casting. (What's happened there is that she's essentially 'cut out' the space between herself and the ground. She won't take more damage for falling farther and faster, but she'll still go 'splat' with whatever velocity she's already picked up.)

Well, our battered, waterlogged mage seems to be going back to her keep - on horseback, I see. I guess that means we're done this time around.