"Velocity is in units of m/s and its velocity at any time during the fall is given by v=at where a is the acceleration of gravity (9.8 m/s^2) and t is the number of seconds. See how the s terms cancel out?"

Actually, no, I don't see ANYTHING cancelled out of the equation... Probably 'cause you didn't show your work. Methinks you'd get a c- from any algebra teacher for that. Hehe...

THE S UNIT OF THE TIME TERM CANCELS OUT ONE OF THE S TERMS IN THE DENOMINATOR OF THE ACCELERATION TERM. THE VELOCITY TERM COMES OUT IN M/S, WHICH IS EXACTLY WHAT YOU'D EXPECT. I DIDN'T THINK IT WOULD BE NECESSARY TO EXPAND IT COMPLETELY.

"As the jug falls, its mass stays the same, but it's velocity is increasing so we describe the jug with a new quantity called the momentum."

This part I wonder about, particularly, the mass staying the same... If it's being accelerated, isn't energy being imparted from the gravitational field. IE, work is being done on the jug to increase or change its velocity, which consumes energy (even if only nominal amounts). Doesn't conservation mean that the energy has to go somewhere, IE into the jug in the form of, what, kinetic energy? And Energy = Mass * C^2. So, it seems to me that as the jug is accelerated (imparted energy) it would increase in mass (even if only a nominal amount), yes?

IN NEWTONIAN MECHANICS, MASS IS DEFINED TO BE THAT PROPERTY OF MATTER WHICH RESISTS ACCELLERATION. IT DOES NOT INCREASE WITH THE VELOCITY OF THE MASS. iN RELATIVISTIC MECHANICS, THE MASS DOES INCREASE WITH VELOCITY, BUT THIS IS NOT MEASURABLE AT THE NORMAL SPEEDS OF OUR EVERYDAY EXPERIENCE, AND YOU MUST FULLY UNDERSTAND NEWTONIAN PHYSICS BEFORE YOU GET INTO RELATIVITY. THAT'S WHY I DIDN'T BRING IT UP. DON'T MIX UP NEWTON'S LAWS WITH YOUR SUNDAY-SUPPLEMENT RELATIVITY OR YOU WILL WIND UP TWISTED IN CONCEPTUAL KNOTS. LEARN TO CRAWL BEFORE YOU TRY WALKING AND YOU WILL EVENTUALLY LEARN THAT NEWTON AND EINSTEIN ARE BEAUTIFULLY RECONCILED.

Well, the fact that it has a name, is neither here nor there. As far as I'm concerned, naming is of no relevance. We could call Kilogram-meters "Bobs" or "Susans" and it still wouldn't make them make any more sense... The naming has nothing to do with what's actually being described. One of those language-y things that has no relevance to the actual maths, as you so nicely pointed out. Language is imperfect. Language is an approximation. Even math is an approximation to the best of our ability.

THAT'S EXACTLY THE POINT I WAS TRYING TO MAKE. SO WHAT'S YOUR POINT?

Technically, if one wanted to get silly one could even say that since math is a language, science-fiction can be written in math. IE, complete gibberish can be written in math that doesn't accurately decribe OUR reality, but only some hypothetical universe (frankly, I think that divorcing physics from observation and saying that only pure math can describe the universe is silly; any language can be written well or written poorly based on the thought processes that go into it, math is no better or worse than any other language that way, and since it's often very abstract it may end up describing nothing useful even if both sides of the equals sign match up).

I NEVER SAID THAT. PHYSICS IS ALL ABOUT OBSERVATION. MATH IS JUST FOR THE BOOKEEPING

And again, isn't the above example/equation you mentioned a problematic statement (exemplary of what NOT to do and how silly errors lead to bad data)? I think you cancelled one too many terms? 'Cause we're multiplying a mass times an acceleration. Where did the mass go in the final number you proferred? Mass doesn't cancel with meters/second^2... Shouldn't that be 39.2 Kg-m/s^2? Congratualtions. You've just written some good science fiction in "math."

YOU'RE ABSOLUTELY CORRECT, I MADE A MISTAKE AND LEFT OUT THE MASS TERM. THE CORRECT FINAL EXPRESSION SHOULD BE
"we know the force of gravity on that jug is a constant 39.2 KG-m/s^2, or 39.2 Newtons". I APOLOGIZE FOR THE TYPOGRAPHICAL ERROR.

On an aside, we still seem to not really understand what a mass-meter is (just like in momentum; except this time we have a unit for it; whooppee! We named something we don't understand! Can't we call kilogram-meters Bobs? C'mon. Pretty please?). And this "term" still seems slightly incomprehensible. I mean what does mass times distance really MEAN? And divided by time? Heck, even mass over time...

SPARE ME THE SARCASM, SMARTASS, OR YOU CAN GET YOURSELF ANOTHER TUTOR. THIS IS WORK FOR ME AND I'M NOT CHARGING YOU FOR IT.

UNITS ARE DEFINED FOR PHYSICAL QUANTITIES BECAUSE THEY SHOW US HOW TO CORRECTLY COMPUTE WITH THEM. WE SAY MOMENTUM IS A NUMBER ASSOCIATED WITH A CERTAIN MASS TRAVELING AT A CERTAIN SPEED. WE CALCULATE THE MOMENTUM BY MULTIPLYING THE MASS TIMES THE VELOCITY (I.E., A 0.05 KG BULLET TRAVELING AT A 1000 M/S HAS A MOMENTUM OF 50 KILOGRAM-METERS PER SECOND. THE MOMENTUM UNIT DOES NOT HAVE A NAME HONORING SOME DEAD PHYSICIST, BUT THAT IS JUST HISTORY AND TRADITION, NOT PHYSICS.
THE REASON WE DO THIS IS SO WE CAN USE THIS UNIT IN COMBINATION WITH OTHER QUANTITIES. FOR EXAMPLE, ONE DEFINITION OF FORCE IS "THE TIME RATE OF CHANGE OF MOMENTUM". SO IF YOU DIVIDE MOMENTUM BY TIME, YOU GET FORCE, SINCE THE DEFINITION OF FORCE IS KILOGRAM METERS PER SECOND SQUARED. ANOTHER DEFINITION OF FORCE, MASS TIMES ACCELLERATION ALSO COMES OUT IN THE SAME UNITS.

PHYSICISTS DID NOT DO THIS ARBITRARILY, BECAUSE IT SOUNDED NEAT. AFTER MANY EXPERIMENTS IN THE REAL WORLD, THEY FOUND THAT CERTAIN MEASUREMENTS COULD BE PREDICTED IF YOU DEFINED THEM A CERTAIN WAY, AND THAT THE WAY THEY VARIED (OR DIDN'T) COULD LEAD TO GREAT INSIGHTS.

YOUR CONFUSION ABOUT UNITS IS A PROBLEM EVERY FIRST-YEAR PHYSICS STUDENT HAS TO COME TO GRIPS WITH, SINCE IT DOESN'T MAKE SENSE THAT SUCH DIFFERENT PHYSICAL PROPERTIES ARE CONNECTED. JUST ACCEPT IT AS A BOOKEEPING CONVENTION. THE PROOF THAT IT WORKS COMES FROM THE LABORATORY, WHEN YOU START SEEING HOW THESE PROPERTIES CAN BE MEASURED AND THEN PREDICTED.

So, I was wondering what does multiplication mean in terms of distance and mass. But then I remembered the rule that division and multiplication can be interchanged by flipping the thing we're working with. IE, rather than multiplying by D, we could divide by 1/D. And we know that division means taking whatever we're dividing and dividing it over a specific area. Then it becomes almost more like an area or volume measurement. IE, some unit, per some other unit... I mean, heck, if we're playing with math, why not, right? Again, I don't know what it means exactly, but we could certainly rephrase the equations:

Force = Kg * D/(S^2)
Force = Kg * D * 1/(S^2)
Force = Kg * 1/(1/D) * 1/(S^2)
Force = Kg/(1/D) * 1/(S^2)

YES, ALL THOSE STATEMENTS ARE PHYSICALLY EQUIVALENT BECAUSE THEY ARE MATHEMATICALLY EQUIVALENT. WHAT IT MEANS IS THAT IF A MASS OF X KG IS ACCELLERATING AT Y METERS PER SECOND PER SECOND IT MUST HAVE A FORCE ACTING ON IT OF XY NEWTONS. OR TO GO BACK TO OUR EXAMPLE, A FOUR KILOGRAM MILK JUG FALLING IN EARTH'S GRAVITATIONAL FIELD OF 9.8 METERS PER SECOND PER SECOND WILL HAVE A FORCE ACTING ON IT OF 39.2 NEWTONS. AND A NEWTON IS DEFINED IN UNITS OF KG-M/S^2, OR THE UNITS OF MASS MULTIPLIED BY THE UNITS OF ACCELLERATION.

But, somehow it seems to me to make more sense if we're saying something akin to, the longer it drops, the more energy is imparted, the more momentum changes, and the more mass it gains due to work performed on it, and conservation laws. Is that a fair assessment? Maybe I'm just having trouble with the math of it. But that seems to me to be what's going on.

YES, WHAT YOU JUST SAID IS ABSOLUTELY CORRECT. BUT WITH THE MATH AND THE UNIT DEFINITIONS YOU CAN DO PRECISE CALCULATIONS WHICH CAN BE TESTED IN THE LAB. YOU CAN'T DO THAT WITH JUST STATEMENTS IN ENGLISH (OR LATIN, WHICH IS HOW NEWTON WROTE IT).

IE, the more energy something has, the more mass it gains. By doing work on something, energy (thus mass) is imparted increasing velocity (even if only nominally). So, an object accelerated in a gravitational field gains mass. Yes?

Now, if an object were imparted energy in such a way as to give it spin, would it still gain mass? IE, if work is done on it, to cause it to rotate faster and faster, does it gain mass?

THIS MAY BE TRUE IN A RELATIVISTIC ENVIRONMENT, BUT THAT'S NOT WHAT WE'RE DISCUSSING HERE. THE MASS OF AN OBJECT IS NOT AFFECTED BY ITS MOTION UNTIL YOU GET TO COLOSSAL VELOCITIES, AND THE RELATIVISTIC EQUATIONS NEEDED FOR THAT ARE BASED ON NEWTON'S LAWS. YOU CAN'T TALK ABOUT RELATIVITY UNTIL YOU HAVE MASTERED CLASSICAL MECHANICS. PLEASE, PLEASE, DON'T MIX THE TWO OR YOU COME UP WITH NONSENSE.

AGAIN, THIS CAN'T BE DONE OVER THE NET, FIND A PHYSICIST OR ENGINEER TO GUIDE YOU THROUGH THIS, OR GO TO A BOOKSTORE AND BUY A 1ST YEAR NON-CALCULUS COLLEGE PHYSICS TEXTBOOK. YOU CAN'T FIGURE OUT HOW PHYSICS WORKS BY JUST THINKING ABOUT IT...UNLESS YOU'RE ISAAC NEWTON.