Newtons Laws -
1. objects tend to keep whatever state of motion they have, unless an unbalanced (ie net) force (like friction) acts on them
2. objects tend to keep the motion they have as a factor of there Mass.
Force = Mass x Acceleration or simply F=mA
both force and Acceleration are vectors (have magnitude and direction), and thus the acceleration (change in motion) will only be in the direction of the net force.
3. every force has an equal and opposite force. i push on the wall, the wall pushes back on me.
Work, Energy and Power
Energy and work have a really circular definition. energy is the ability to do work (called joules), and work is defined as the changing of an objects energy(also measured in joules). Power is defined as the amount of work in a time interval(joules per second). the English units typically used are ft*lbsf (called foot pounds) and ft*lbsf/sec. 550 ft*lbsf/sec = 1 hp.
Work = Force x Distance (the force must actually move the object to do work)
kinetic Energy = (1/2)mv^2
Gravitational Potential Energy = ghm
so, an example of energy and work is simply lifting a stone.
a stone on the floor unmoving has no mechanical energy. when i lift the stone, i am doing work on it (applying a force and moving it a distance). so now the stone has more energy. the time it took me to raise the stone is how much power i have (the faster the more powerful).
Conservation of Energy
this is a fundamental tenant of the universe. under some circmstances it can be broken (near the speed of light...), but then we are talking nuclear mechanics. NOTHING like has ever or will ever take place in a paintball gun. for all realistic modeling of paintball and anything related to it, this is a law harder than a 17 year old who took 5 Viagra.
ENERGY MAY NOT BE CREATED OR DESTROYED, IT CAN ONLY CHANGE FORMS.
the perfect example of this is the coaster example. in a roller coaster, you build up a tremendous amounts of gravitational potential energy, and though the course of the ride you trade your potential energy to kinetic energy and back again maybe a dozen times.
of course eventually the coaster will come to a stop right? doesn't that seem like energy is being destroyed?
gravity is the force converting all potential energy to kinetic, and then it is doing negative work to the coaster when it takes the potential back from the kinetic. gravity is doing alot of work. however, another force is doing alot of work too - friction. friction is doing work (transferring energy) from the coaster the whole time. friction is converting kinetic energy to heat energy. another force is also doing work - air resistance. air resistance is transferring kinetic energy into kinetic energy of air. in fact, one might even say that the coaster is doing work on the air around it, to move it.
all this talk of friction means i should probably define it for you. friction is caused by the molecules of one object being attracted to the molecules of another object. think of it like velcro, hooks on one side, fuzzy stuff on the other. now, what causes friction is somewhat minor when we talk about mechanics because we simply dont care. friction is real, and thats really all that matters.
there are two types of friction, kinetic, and static friction. kinetic friction is between two bodies that are moving relative to each other, and static is between two bodies that are stationary to each other.
to calculate friction, we need to use something called the co-efficient of friction. this a a property value that we look up in a book. obviously since different types of objects have different amounts of friction its easy to see that the surfaces pushed together and what they are made of is important.
Materials in Contact........Coefficient of Static Friction* S........Coefficient of Kinetic Friction * K
Wood on wood................................0.5.....................................0.3
Waxed ski on snow..........................0.1 .......................................0.05
Ice on ice .......................................0.1................................0.03
Rubber on concrete (dry)..................1.0.......................................0.8
Rubber on concrete (wet).................0.7........................................0.5
Teflon on Teflon .............................0.04......................................0.04
Synovial joints (in humans) ..............0.01.....................................0.01
* These values are approximate and intended only for comparison.
now, another thing you can see is that static friction is ALWAYS greater than or equal to kinetic friction. that makes sense, it explains why you can do break stands in a cars and such. it explains why drifting and burning out are bad, and why indy cars and drag racers never do that.
so the equation for calculating friction is - Force of friction = Coefficient of friction x the force pushing the two surfaces together
usually shortened to F = (mu)N
notice, surface area is nowhere in that equation. the size of the surfaces being pushed together does not matter when calculating friction.
Air Resistance and other damping forces
air resistance is a pickle. see air resistance is a differential relationship. meaning, you need to know calculus to solve most fluid resistance problems. i cannot possibly teach you calculus but i will say a bit about this.
the reason it requires calculus is because air resistance is a function of velocity. velocity is the derivative of the position function, so theirs where you get your calculus.
but one thing i can tell on relatively simple terms is that the faster an object is moving (velocity) the larger the force. often times its modeled as a power function, meaning the resistance is equal to the velocity squared. other inputs into the force of air resistance are shape and surface texture.
fortunately in paintball, most of the time shape, mass and all that jazz is a constant when comparing things because most paintball are pretty similar.
The Calculus of Motion or kinematics (sort of)
ok, i figured since i mentioned velocity and that whole bag i would explain a bit about that. Sir Issac Newton in england and Libneitz (sp?) in germany both discovered calculus almost simultaneously because both were studying motion and realized how important rate of change is to the study of motion. they are both given credit for the discovery.
anyway, the important thing is the rate of change of an object.
the position function describes the path an object takes, and you can use it to find where the object is at any given time.
if we take the derivative (otherwise known as finding the rate of change) of the position, we get velocity. velocity is the rate of change of the position. velocity is alot like the speed an object has, however speed is simply a number, and velocity has a direction and a magnitude. it is enough to say that the velocity is how the position of the object is changing, how fast, and in which direction is the position changing.
if we take the derivative of the velocity, the rate of change of the velocity, we will have the acceleration. acceleration describes how the velocity is changing, and in which direction. the acceleration is very important because of the previously mentioned F = mA. Acceleration, like velocity is a vector, and has both a value and a direction.
something we deal with ALL the time in paintball. pressure is simply a Force over an Area. whenever we have a force that is pushing against an area, we have pressure. no, pressure does no only apply to fluids but to anything that has a force and an area.
the ideal gas law is the start of ideal fluids. there are about a million and half differnet versions of the ideal gas law putting it in different terms, like molar mass, mass, and whatnot. the basic chemestry form is
PV = nRT
where P is pressure, V is volume, and T is absolute temperature. R is a constant, and n is the molar mass (if i remember right)
basically, as long as your working with a fixed amount of mass (like we typically are in paintball), pressure and volume will go up if temperature rises, and if temp remains the same, then pressure and volume must trade off.
ideal fluid flow
the only real way to understand fluid mechanics is to use energy. as stated before, energy is neither created, nor destroyed. so the energy at the start of your dynamic system must be equal to the energy at the end. so fluids have many different types of energy -
Pressure is a measure of potential energy (P)
Velocity is a measure of kinetic energy (V)
and gravitational potential energy (z)
later, i will discuss thermodynamic properties of fluids. in most cases in paintball, heat and that energy is just not a very large factor, so we can neglect it. in most cases we neglect the gravitational effect aswell because air is not very heavy, and its not moving in that direction very much.
what those three terms build for us is the Bernoulli equation. again, it has a few different forms but most commonly stated -
(P/density*g)+(V^2/2*g)+z = constant throughout the system.
in this equation, the first term is pressure potential energy, the second is kinetic energy, and the third is the gravitational potential energy. every term in this equation translates directly into feet, or meters. in fluids we call feet of elevation (what it represents), head. yeah, i know. funny stuff right?
anyway, we use this equation to solve for all sorts of different stuff. if we know the pressure in a dump chamber (velocity of the fluid is zero, and z term is neglected) and we vent it, we can predict how fast the fluid will be moving (pressure drops to zero, z term still neglected) because we know that energy is constant, it just changes form.
another useful equation in basic fluid mechanics is for solving flow. the letter Q is often used.
flow is defined as the amount of stuff going though a specific area. the equation is simply Q = Area(V)
this again, must be constant (assuming no leaks). so basically it means that if the cross sectional area shrinks, the velocity must be faster.
Stats and Experimental Design
i really hate stats. unfortunately, knowing the basics of stats is HUGELY important in scientific testing.
the reason we use confidence intervals and other stats methods to look at our data (deals with gun consistency) -
the issue is finding the "true mean" vs the "sample mean" if i want to find the average age of all poeple in the entire world, what do i do?i decide on what would be a representative sample of the world, poll those poeple, and calculate the mean right?
when we do that we are finding a sample mean.
the question is - how close to the true mean (if you actually polled everyone on the entire planet) age?
that's where the CI comes in.
so yes, we can do a comparative study by agreeing on a common sample size and say "well that guns three shots are more consistent than that guns three shots"
however, what we really want to know is how consistent EVERY shot from the gun is.
to do this, we are comparing a sample mean (the shots fired over the chrono) to the true mean (the velocity of ALL shots the gun ever fires).
a CI simple compares the variation in the data (standard deviation), with the number of samples taken, and then shows you how accurate to the true mean our sample mean is.
so you see, we use confidence intervals to look at how different our sampled mean (the data points we take in a test) to the true mean (if we had data on every single event under those conditions).
the equation for a two sided confidence interval is (mean) +/- (tvalue x standard deviation)/ Sqroot(number of events)
the t value is a textbook value based on a normal distribution, and we can decide how inclusive our results are by judging if we want to 90%, 95%, 99%, or even 99.9%. what that means that if we take a sample size of say muzzle velocities, then calculate the CI around it, we choose how inclusive our range is. if we calculate the CI based off a 90% t value, then we can be 90% sure that our next group of samples of velocity will fall in that range. with a CI we can look at exactly how good our sample is compared to the "true" values of ALL shot velocities.
MORE to come. comments please?