Thursday, March 17, 2016

Nymphs Part 2: Mountain Nymph

The crystal clear water of the icy pool shimmered and rippled in the still air of the morning. Gradually, the ripples grew in an upward undulation that formed shapely calves, then thighs, and eventually a fully formed woman of surpassing beauty.
Nymphs are loci geniuses that inhabit various natural places. While similar to what are sometimes called “fae”, they more closely resemble elementals of a specific locale. They are inherently tied to the location they inhabit; indeed, their very existence depends on the wellbeing of the environment. And they defend it vehemently.

All nymphs exercise a measure of control over their homes and use this, in combination with their preternatural beauty to ward off attacks. Their beauty is terrible and captivating, and they use this mercilessly to their advantage.

As dangerous as nymphs can be, they are excellent sources of information on the area in which they dwell, and if befriended, they make powerful allies. However, their legendary shyness makes this difficult, and their sadistic impulses make it doubly dangerous!

Mountain Nymph

CER 73 (OR 27 and PR 46)
Mountain nymphs are spirits of place who watch over mountainous areas.  They are closely attached to particular passes, grottos, peaks, etc., and may not stray more than a short distance from them without becoming seriously ill.  They defend their homes fiercely, employing magic to animate the mountain itself, often while dancing about high above.  All mountain nymphs command great knowledge about their environment.
ST: 9                                                       HP: 10                                                    Speed: 6.00
DX: 12                                                    Will: 10                                                  Ground Move: 6
IQ: 10                                                     Per: 11                                                   Climbing Move: 6
HT: 12                                                    FP: 12                                                     SM: 0

Dodge: 9                                                Parry: 10 (Unarmed)                           DR: 5

Punch (14): 1d-1 crushing. Reach C.
Quarterstaff (14): 1d+1 crushing (swing) or 1d crushing (thrust). Reach 1, 2*.
Animated Environment: Mountain nymphs can animate the environment around them to slide or knock intruders from their feet (target takes 3d crushing double knockback no wounding), strike at them (1d crushing or impaling) with rocks from above, or cause the ground to partially swallow a person – treat as Binding 10.
Difficult Terrain: A mountain nymph can cause anyone within their domain to have difficulty maintaining their balance or moving about.  The victim must roll a Quick Contest between his Will and the nymph's Will.  On a failure, the subject suffers a -2 penalty to DX and -2 penalty to Acrobatics, Jumping, Climbing, and Skiing lasting a number of minutes equal to his margin of failure.
Stunning Beauty: By striking even the slightest pose, anyone who sees her must make a Fright Check at -5 and roll on the Awe table.
Threshold Entity: This being doesn’t breathe, drink, eat, or sleep, is immune to metabolic hazards, and is either insubstantial and invisible or substantial and visible.

Traits: Acute Hearing 2; Appearance (Very Beautiful; Universal); Berserk (9) (Special Trigger, harming the wilderness); Callous; Can Be Turned by True Faith; Curious (15); Dependency (Home Mountain; Daily); Dislikes Loud Noises; Divine Curse (Keep to the letter of any promise); Fearlessness 2; Higher Purpose (Protect Home Mountain); Impulsive (12); Odious Personal Habit -1 (Capricious); Perfect Balance; Sadism (15); Sense of Duty (Home Mountain); Shyness (Mild); Super Jump 1; Terrain Adaptation (Mountain); Unaging; Vulnerability (Iron x2).
Features: Affected as Spirit.
Skills: Area Knowledge (Local)-12; Brawling-14; Climbing-16; Dancing-16; Intimidation-17; Jumping-16; Naturalist-12; Quarterstaff-14; Sex Appeal-12; Stealth-14.
Notes: Most nymphs won't negotiate because they are too shy.  Those who do are amiable enough, but dealing with such an astounding beauty is often disconcerting – especially while under the effect of her intoxicating beauty (see above).  Always scars from wounds inflicted by iron.

Monday, March 14, 2016

This Time It IS Rocket Science!

This time, I’m going to get into how I make rough apogee predictions for model rockets. Below is a derivation of the formulas necessary for predicting the apogee of a spherical bovine rocket undergoing air drag.

Warning: Math Ahead

So ultimately, we want to know how high the rocket will go. I am going to call this “y”, “displacement” or “apogee” interchangeably. I might even call it the height. Either way, this is what we are ultimately trying to solve for.

Anyone who has taken a basic calculus or physics class probably recognizes a few truths:
That is to say, the change in location of an object over a period of time is its velocity, and its change in velocity over a period of time is its acceleration. Thankfully, we can also say this in reverse:
This becomes important when we consider that the flight of any object is based on the sum of the forces acting on it. This is grounded in Newton’s Laws of Motion. Thus we know that the sum of our forces is

where F = force, m = mass, and a = acceleration. Thus,

Now we have a few terms here that we need to sort out: FThrust, FDrag, and FWeight. Of these, we can generally not worry about FThrust because that information is available based on our motor. Likewise, we know that FWeight = mg, where g is the acceleration due to gravity. This leaves FDrag for us to define:

where ρ = the density of air, CD = the coefficient of drag, A = the crossectional area of the rocket, and v = the velocity. This may seem like a lot of information, but much of it really doesn’t change with time for our purposes here. So I am going to combine a bunch of these into a constant. I will do this a few more times to clean up equations and put them in more easily recognizable forms, too.

Now we have a simpler looking equation:
Substituting this and the formula for FWeight, we get:
One more substitution, referring back to the beginning gives us
and we are ready to separate the variables and integrate.
But first, we are going to use some algebraic manipulations to get this looking like something we can integrate more easily:
Now it’s time to integrate both sides:
Both m and k are constants, so we pull those out and integrate the right side:
We should now recognize the integral on the left as tanh-1 u:
By the definition of tanh-1, we can perform the following manipulations:
And define the constant
This lets us say:
Raising both sides to the e power gives:
Then we multiply both sides by the denominator:
Expanding the right side gives us:
Next, I need to isolate my v’s on one side:
Next, factor out v:
And divide both sides by (-ext-1):
Put more neatly:
And thankfully, we know that t = impulse divided by thrust, both of which are specifications provided by our motors. So now we know the final velocity when the motor finishes burning its fuel, but that doesn’t tell us either how high the rocket is at that point or how far it continues to coast. For that information, we need more integration.

We will begin by solving for yBurn, or how far the rocket travels while the motor is still providing thrust. For simplicity’s sake, we will assume the rocket’s weight does not change during this time (we are launching a frictionless spherical cow, aren’t we?), so m will continue to refer to the mass of the rocket while it is full of propellant.

Referring back to our original equation, we know that:
So we can say
By separating the variables and integrating both sides, we get:
Which becomes:
Now we know the rocket’s altitude when the motor cuts out, but the rocket won’t instantly stop; it will continue to coast. So let yCoast = the distance the rocket coasts on its momentum before beginning its descent. That is to say, how far will the rocket travel upward once FThrust = 0. Note that since the rocket has exhausted all of its fuel, we no longer use mFull – we are now use mEmpty.
This should all look a bit familiar, but not that because FThrust = 0, that term is simply missing:
By integrating both sides, we get:
Which becomes:
Now, we know the rocket’s apogee will be the sum of how high it goes while under power (yBurn) and how far it coasts before the acceleration due to gravity starts returning it to Earth (yCoast):
And there you have it! To summarize:

A = the area of the rocket’s cross-section normal to its velocity vector.
CD = the rocket’s coefficient of drag.
FThrust = the motor’s thrust in Newtons.
g = the acceleration due to gravity.
I = the motor’s impulse in seconds.
mEmpty = the mass of the rocket after burnout.
mFull = the mass of the rocket at liftoff.
ρ = the density of air.

Of course, the astute reader will notice that through all of that mess, there is one term still undefined- CD. This is the coefficient of drag, and determining it generally requires experimentation or running simulations. For my purposes, that is where OpenRocket and SolidWorks come in. Both are capable of determining CD.

Did you survive? I warned you there’d be math!

Monday, March 7, 2016

It's Not Rocket Science!

Today's post was going to be a derivation of the formulas I used to predict a rocket's performance. It does get pretty math-intensive, and so I wrote it out ahead of time in word to more easily proof before pasting into here. This was a mistake. The formulas didn't transfer. Not only that, but I'm still looking for a good way to put them and any future math I may decide to include, in this blog. So instead of math, you're getting a llama.

Was this better than a few pages of integration?