1. Momentum in a closed system is neither created nor destroyed.
Let's start by looking at those terms.
Momentum is the mass (m) of an object times its velocity (v). That's
just how it's defined. Frankly, momentum would be a pretty useless
thing to know, except that it is conserved. That makes it important,
as we'll see.
A closed system is one where nothing's going in or coming out. Three
moving pool balls on a frictionless pool table can be a closed system.
They can bounce off the pool table sides and off of each other, forever
and ever. If I reach in and grab one of the balls to stop it, I've
just intruded on that closed system. The Law of Conservation of Momentum
won't apply then.
"Neither created nor destroyed" means that the momentum of a system
has to be the same at all points in time.
Two other things we'll need to know: A perfectly elastic collision
is one where two or more items bounce off of each other with no losses
to friction, deformation, or anything. A perfectly inelastic collision
is nearly the opposite: two or more items collide and stick to each
other, with no losses.
OK. So what?
So, all in all, momentum isn't a terribly useful concept unless you're
analyzing collisions of billiard balls or cars or something. But there
are two main uses for this knowledge I can think of off the bat. The
first relates to the Levitate spell. The other relates to out-of-game
arguments and how to judge them. We'll do the easy one first.
Levitation
Say your PCs have just figured out that Levitate makes a nifty attack
form. In fact, they've just levitated one of your villains up into
the air and left him there. There's a tree nearby, just out of reach.
If he could only get to it, he could pull himself around a little.
So the villain takes his boot, or dagger, or something, and throws
it in the opposite direction of the tree. Then he'll slowly glide
over.
How's that again?
Well, to start with, the villain and his dagger have a mass M_total
= M_villain + M_dagger, and neither is going anywhere, so their initial
velocity, v_i, is zero. Their momentum is:
M_total * v_i = M_total * 0 = 0
Zero. Doesn't look useful, does it? Well, now consider the villain
throwing the dagger away. Because momentum is conserved, the total
momentum of the elements of the system - that is, the villain and
his dagger - must always equal zero, since that's what we just figured
out it equaled to start with! If the villain moves at v_villain and
throws the dagger at a speed v_dagger:
(M_villain * v_villain) + (M_dagger * v_dagger) = M_total * v_i =
0
Or:
M_villain * v_villain = -M_dagger * v_dagger
Numbers! Numbers, you say! All right. Let's say the villain has a
mass of 90 kg (about 200 lbs), and the dagger has a mass of about
1 kg (about 1/2 lbs.) And the villain's got a pretty good throwing
arm, so he whips the dagger out at about 15 m/s (around 5 yards/second).
How fast does the villain go?
90 kg * v_villain = -1 kg * 15 m/s
The villain travels at a whopping 1/6 m/s (about 6 inches/sec), with
the negative sign meaning it's in the opposite direction as the dagger.
Not too fast - but he can travel around 10 yards in one round. That
ought to be plenty to get him to the nearest tree or cliff, so he
can grab something and maneuver a bit.
There's another way to do this, too: The villain could ask someone
on the ground to throw him something. It can either bounce off of
him (an elastic collision) or he can try to catch it (an inelastic
collision) - in either case, he'll get some momentum out of the deal.
For the perfectly elastic case, all the thrown object's momentum will
go into the villain, so the object's (horizontal) velocity will be
zero. It'll fall straight down to the ground. You'll have to decide
how fast the object was thrown. In this case:
(M_object * v_object_start) + (M_villain * 0) = (M_object * 0) + (M_villain
* v_villain_end)
The zero on the left hand side is the villain's starting velocity
(zero - he's just hanging there), and the zero on the right hand side
is the object's ending velocity (also zero, as discussed above.) If
someone bounced a 1 kg rock thrown at 15 m/s off of him, the villain
would end up moving at 1/6 m/s, just like the case above.
What if he caught the rock? Then he'll move a little more slowly:
(M_object * v_object_start) + (M_villain * 0) = [(M_villain + M_object)
* v_both_end]
If you put in the numbers and do the math, you'll see that the villain
and rock together will drift off at 0.165 m/s, which is a tiny bit
slower than 1/6 = 0.167 m/s. And then he can throw the rock away and
give himself another momentum kick!
Arguing: Force, Energy, and Momentum
Time and again, I've seen the concepts of force, energy, and momentum
confused and abused around the gaming table. Here are some definitions
for them, both in plain English and in physics equations:
Force: A push or a pull. Force = mass * acceleration
Energy: The ability to do work. Kinetic energy = 0.5 * mass * velocity^2
Momentum: No good intuitive definition. Momentum = mass * velocity
Acceleration: Change in velocity over time. Acceleration = (end_velocity
- start_velocity)/(end_time - start_time)
Velocity: Change is position over time. Velocity = (end_position -
start_position)/(end_time - start_time)
Let's say you're charging a shield wall. You might say that you're
"gathering momentum" with your charge. Well, you are, but that's not
really relevant. Having momentum doesn't guarantee that you'll have
the sort of collision you want with your object. In a worst case scenario,
you could smack into the shield wall and bounce off in a perfectly
elastic collision that sends you clear back to where you can from!
You had momentum - but it didn't do you any good.
What you're really gathering is energy and, in a weird way, force.
When you crash into your opponent, you transfer your energy to him.
Sudden transfer of kinetic energy to a biological system hurts! That's
what bullets do, after all. You also apply a force. You're moving
at some high speed when you're charging and contact with the enemy
slows you down to nearly zero forward speed. Your speed changed -
so you accelerated. (We often call negatively accelerating decelerating,
if you like that term better). You decelerated, so you applied a force.
The faster you were going, the more you'll slow down, and the greater
your deceleration - and thus, the greater your force. By Netwon's
Third Law, your opponent must generate an equal and opposite force,
or else be pushed aside. You hope that by charging fast enough, you'll
generate enough energy to hurt him badly and enough force to push
him aside. Momentum has nothing to do with it, really.
Notice: Kinetic energy increases directly with mass. This is why big
guys with hands like cooked hams can hurt people easily - they strike
with a lot of mass behind the blow.
Also notice: Kinetic energy increases directly with the velocity squared.
As you go faster, you can apply much, much more energy. This is why
small, fast guys in karate class can hit so hard. It's not that they're
thumping you with huge fists, it's that they're going so fast they
generate more energy.
It's a lot, isn't it? It can be very confusing, trying to sort out
the velocities and the accelerations and the forces and the energies.
Which does what and to whom? Like I said in the Intro: Physics and
gaming don't generally mix, especially if you don't have a very solid
understanding of the physics. It's very easy to get caught in imprecise
defintions of words that have very precise meanings. Luckily, you
have a fairly good intuition about these things: you apply forces
every day, after all. Wing it, make a ruling, and if your PCs want
to drag out their physics textbooks to argue with you, make them wait
until after the game is over.