Aristotle’s laws of motion
Aristotle (Ἀριστοτέλης) 384–322 BCE was a Greek philosopher and scientist born in the city of Stagira, in classical Greece.
At seventeen years of age, he joined Plato’s Academy in Athens and remained there until the age of thirtyseven (c. 347 BC).
His writings cover many subjects – including physics, biology, zoology, logic, ethics, poetry, theater, music, linguistics, and politics. They constitute the first comprehensive system of Western philosophy.
Shortly after Plato died, Aristotle left Athens and, at the request of Philip of Macedon, tutored Alexander the Great beginning in 343 BC.
Aristotle’s views on physical science profoundly shaped medieval scholarship. Their influence extended from Late Antiquity and the Early Middle Ages into the Renaissance, and were not replaced systematically until the Enlightenment and theories such as classical mechanics.
 excerpted and adapted from Aristotle. (2016, October 20). In Wikipedia, The Free Encyclopedia.
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Aristotle’s laws of motion.
Excerpted from a lecture by Professor Michael Fowler, U. Va. Physics, 9/3/2008
http://galileoandeinstein.physics.virginia.edu/lectures/aristot2.html
What Aristotle achieved in those years in Athens was to begin a school of organized scientific inquiry on a scale far exceeding anything that had gone before. He first clearly defined what was scientific knowledge, and why it should be sought. In other words, he singlehandedly invented science as the collective, organized enterprise it is today. Plato’s Academy had the equivalent of a university mathematics department, Aristotle had the first science department, truly excellent in biology, but, as we shall see, a little weak in physics.
After Aristotle, there was no comparable professional science enterprise for over 2,000 years, and his work was of such quality that it was accepted by all, and had long been a part of the official orthodoxy of the Christian Church 2,000 years later. This was unfortunate, because when Galileo questioned some of the assertions concerning simple physics, he quickly found himself in serious trouble with the Church.
Aristotle’s method of investigation:
defining the subject matter
considering the difficulties involved, by reviewing the generally accepted views on the subject, and suggestions of earlier writers
presenting his own arguments and solutions
This is the pattern modern research papers follow, Aristotle was laying down the standard professional approach to scientific research.
Aristotle often refuted an opposing argument by showing that it led to an absurd conclusion, this is called reductio ad absurdum (reducing something to absurdity). As we shall see later, Galileo used exactly this kind of argument against Aristotle himself, to the great annoyance of Aristotelians [people who fully agreed with Aristotle] 2,000 years after Aristotle.
[Aristotle himself likely would not have minded later thinkers disagreeing with him; in his lifetime Aristotle would change his mind, if he found new information or a more logical argument.]
In contrast to Plato, who felt the only worthwhile science to be the contemplation of abstract forms, Aristotle practiced detailed observation and dissection of plants and animals, to try to understand how each fitted into the grand scheme of nature, and the importance of the different organs of animals.
It is essential to realize that the world Aristotle saw around him in everyday life was very different indeed from that we see today. Every modern child has since birth seen cars and planes moving around, and soon finds out that these things are not alive, like people and animals. In contrast, most of the motion seen in fourth century Greece was people, animals and birds, all very much alive. This motion all had a purpose, the animal was moving to someplace it would rather be, for some reason, so the motion was directed by the animal’s will.
For Aristotle, this motion was therefore fulfilling the “nature” of the animal, just as its natural growth fulfilled the nature of the animal.
To account for motion of things obviously not alive, such as a stone dropped from the hand, Aristotle extended the concept of the “nature” of something to inanimate matter. He suggested that the motion of such inanimate objects could be understood by postulating that elements tend to seek their natural place in the order of things:
So earth moves downwards most strongly,
water flows downwards too, but not so strongly, since a stone will fall through water.
In contrast, air moves up (bubbles in water),
and fire goes upwards most strongly of all, since it shoots upward through air.
This general theory of how elements move has to be elaborated, of course, when applied to real materials, which are mixtures of elements. He would conclude that wood has both earth and air in it, since it does not sink in water.
Natural Motion and Violent Motion
Things also move because they are pushed. A stone’s natural tendency, if left alone and unsupported, is to fall, but we can lift it, or even throw it through the air.
Aristotle termed such forced motion “violent” motion as opposed to natural motion.
The term “violent” just means that some external force is applied to it.
Aristotle was the first to think quantitatively about the speeds involved in these movements. He made two quantitative assertions about how things fall (natural motion):
Heavier things fall faster, the speed being proportional to the weight.
The speed of fall of a given object depends inversely on the density of the medium it is falling through.
So, for example, the same body will fall twice as fast through a medium of half the density.
Notice that these rules have a certain elegance, an appealing quantitative simplicity. And, if you drop a stone and a piece of paper, it’s clear that the heavier thing does fall faster, and a stone falling through water is definitely slowed down by the water, so the rules at first appear plausible.
The surprising thing is, in view of Aristotle’s painstaking observations of so many things, he didn’t check out these rules in any serious way.
It would not have taken long to find out if half a brick fell at half the speed of a whole brick, for example. Obviously, this was not something he considered important.
From the second assertion above, he concluded that a vacuum cannot exist, because if it did, since it has zero density, all bodies would fall through it at infinite speed which is clearly nonsense.
For violent motion, Aristotle stated that the speed of the moving object was in direct proportion to the applied force.
This means first that if you stop pushing, the object stops moving.
This certainly sounds like a reasonable rule for, say,
pushing a box of books across a carpet, or an ox dragging a plough through a field.
(This intuitively appealing picture, however, fails to take account of
the large frictional force between the box and the carpet.
If you put the box on a sled and pushed it across ice,
it wouldn’t stop when you stop pushing.
Centuries later, Galileo realized the importance of friction in these situations.)
Math is the language of physics
Mathematics is the language of physics
Natural philosophy [i.e., physics] is written in this grand book – I mean the universe – which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. [The universe] cannot be read until we have learned the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.
 Galileo, Opere Il Saggiatore p. 171.
Mathematics is the language of physics. Physical principles and laws, which would take two or even three pages to write in words, can be expressed in a single line using mathematical equations. Such equations, in turn, make physical laws more transparent, interpretation of physical laws easier, and further predictions based on the laws straightforward.
 Mesfin Woldeyohannes, Assistant Professor, Western Carolina University
ἀεὶ ὁ θεὸς γεωμετρεῖ – Aei ho theos geōmetreî. God always geometrizes.
 Plato, 400 BCE, classical Greece, as quoted by Plutarch in his The Moralia, Quaestiones convivales. (circa 100 CE)
The Unreasonable Effectiveness of Mathematics in the Natural Sciences
Wigner begins his paper with the belief, common among those familiar with mathematics, that mathematical concepts have applicability far beyond the context in which they were originally developed. Based on his experience, he says “it is important to point out that the mathematical formulation of the physicist’s often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena.”
He then invokes the fundamental law of gravitation as an example. Originally used to model freely falling bodies on the surface of the earth, this law was extended on the basis of what Wigner terms “very scanty observations” to describe the motion of the planets, where it “has proved accurate beyond all reasonable expectations”.
Another oftcited example is Maxwell’s equations, derived to model the elementary electrical and magnetic phenomena known as of the mid 19th century. These equations also describe radio waves, discovered by David Edward Hughes in 1879, around the time of James Clerk Maxwell’s death. Wigner sums up his argument by saying that “the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it”. He concludes his paper with the same question with which he began:
The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.
 The Unreasonable Effectiveness of Mathematics in the Natural Sciences. (2016, September 11). In Wikipedia, The Free Encyclopedia
The Unreasonable Effectiveness of Mathematics in the Natural Sciences
Scratch Resources
Resources
The Pen is a feature in Scratch that allows a Sprite to draw shapes, plot colored pixels, and so forth on the screen with the Pen Blocks. Lines, dots, rectangles, and circles are the easiest shapes to draw, but with enough scripting, any shape can be created.
https://wiki.scratch.mit.edu/wiki/Pen
A graphic effect is an effect that can be used on a sprite or the Stage, changing their look in some way. These blocks can be found under the Looks section. No effect, color, fisheye, whirl, pixelate, brightness, ghost, mosaic
https://wiki.scratch.mit.edu/wiki/Graphic_Effect
Projects
Plot a line ( y=mx +b) Hit ‘space’ to plot a line and the c key to clear the old lines.
https://scratch.mit.edu/projects/20613546/
Speed=Distance/Time: Doesn’t actually calculate numerical answers. Instead, it changes the size of the S, D and T graphical icons according to their value.
https://scratch.mit.edu/projects/97105344/
Make your own line graph (primitive)
https://scratch.mit.edu/projects/51897174/
Drawing fractals
http://www.edutopia.org/blog/scratchprogrammingadvancedfractalfundylanryder
Advanced projects using physics and math:
Use Trigonometric principles to determine Vectors for X and Y Directions for Sprite(Object) Movement.
Simulate Gravity Physics.
Use Conditional Statements and While Loops (Repeat Until) to determine action within Game.
– Use Variables to store, calculate, and direct movement within Game/Simulation.
http://nebomusic.net/advancedscratch.html
CoderDojo: AthenryMultiple advance Scratch topics
https://coderdojoathenry.org/category/scratchadvanced/
Building a 3D wireframe
https://coderdojoathenry.org/2016/04/17/scratchadvancedgeneralpurposewireframe3dengine/
Galaga classic video game
https://scratch.mit.edu/projects/10076961/
Building a platforming game in Scratch
https://www.scribd.com/doc/131949494/BuildingaplatformgamewithScratch
MIT’s Scratch Part 4: Twenty Webs Sites To Support Scratch And The Itch For Transforming Education
Teachict – 5 projects (7 hours+) for fully functional games.
Teachict.com programming/scratch/scratch_home.htm
Mathematics problems
https://scratch.mit.edu/studios/6423/
Recommended books
Scratch Programming for Teens PDF book free online
Adventures in Coding, Eva Holland and Chris Minnick
Scratch Programming Playground: Learn to Program by Making Cool Games, by Al Sweigart
Scratch 2.0 Sams Teach Yourself in 24 Hours, by Timothy L. Warner
Discovery of conservation of momentum
You can explore this historyoriented lesson by Prof. Michael Fowler.
Momentum, Work and Energy Michael Fowler, U. Va. Physics
In 530 A.D., working in Alexandria, Byzantine philosopher John Philoponus developed a concept of momentum in his commentary to Aristotle’s Physics. Aristotle had claimed that everything that is moving must be kept moving by something. For example, a thrown ball must be kept moving by motions of the air.
Most writers continued to accept Aristotle’s theory until the time of Galileo, but a few were skeptical.
Philoponus pointed out the absurdity in Aristotle’s claim that motion of an object is promoted by the same air that is resisting its passage.
He proposed instead that an impetus was imparted to the object in the act of throwing it.
Ibn Sina (Arabic ابن سینا) (known by his Latinized name, Avicenna) read Philoponus and published his own theory of motion in The Book of Healing in 1020. He agreed that an impetus is imparted to a projectile by the thrower – but unlike Philoponus, who believed that it was temporary, and would decline even in a vacuum – Ibn Sina viewed it as a persistent. He understood that it required external forces – such as air resistance – to dissipate it.
These ideas were refined by European philosophers Peter Olivi and Jean Buridan. Buridan, who in about 1350 was made rector of the University of Paris, referred to impetus as proportional to the weight times the speed.
Like Ibn Sīnā, Buridan held that impetus (momentum) would not go away by itself; it could only dissipate if it encountered air resistance, friction, etc.
René Descartes believed that the total “quantity of motion” in the universe is conserved: quantity of motion = size and speed.
But Descartes didn’t distinguish between mass and volume, so this is not a specific equation.
Leibniz, in his “Discourse on Metaphysics”, gave an experimental argument against Descartes’ idea of “quantity of motion”.
Leibniz dropped blocks of different sizes, different distances.
He found that [size x speed] did not yield a conserved quantity.
The first correct statement of conservation of momentum:
English mathematician John Wallis, 1670
Mechanica sive De Motu, Tractatus Geometricus:
Isaac Newton’s Philosophiæ Naturalis Principia Mathematica, 1687
Defined “quantity of motion”, as “arising from the velocity and quantity of matter conjointly”
> mass x velocity – which identifies it as momentum.
Adapted from “Momentum.” Wikipedia, The Free Encyclopedia. 2 Oct. 2015.
External resources
The cause of motion from Aristotle to Philoponus
The cause of motion Descartes to Newton
Learning Standards
2016 Massachusetts Science and Technology/Engineering Curriculum Framework
HSPS22. Use mathematical representations to show that the total momentum of a system of interacting objects is conserved when there is no net force on the system. Emphasis is on the qualitative meaning of the conservation of momentum and the quantitative understanding of the conservation of linear momentum in
interactions involving elastic and inelastic collisions between two objects in one
dimension.
HSPS23. Apply scientific principles of motion and momentum to design, evaluate, and refine a device that minimizes the force on a macroscopic object during a collision. Clarification Statement: Both qualitative evaluations and algebraic manipulations may be used.
Common Core Math
 CCSS.MATH.CONTENT.7.EE.B.4 Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
 CCSS.MATH.CONTENT.8.EE.C.7 Solve linear equations in one variable
 CCSS.MATH.CONTENT.HSA.SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. (including isolating a variable)
 CCSS.MATH.CONTENT.HSA.CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.
 http://www.corestandards.org/Math/
Inertia and mass
Newton’s first law of motion: Inertia
Every object continues in it’s state of rest,
or of uniform velocity,
as long as no net force acts on it.
If at rest, objects require force to start moving
If moving, objects require a force to stop moving

Animations showing Newton’s Law of Inertia

2 different definitions of mass
a measure of inertia
the quantity of matter
Don’t confuse mass with volume
Here are five cylinders of different metals:
they all are different volumes, yet all are of equal mass.
Lead, copper, brass, zinc, and aluminum.
How is this possible?
Somehow, more matter can be crammed into the same volume with denser materials
Less matter takes up the same volume in lessdense materials
Mass is not weight
Weight is how much a mass is pulled down by gravity.
This girl has the same mass on both worlds, yet her weight varies.
Mass is the quantity of matter in an object.
Weight is the force of gravity on an object.
One kilogram weighs (approximately) 10 Newtons
The gravity of Earth gives a downward acceleration,
g = 9.8 m/s^{2}, to objects.
We often approximate this as g ≅ 10 m/s^{2}
Because the object is being accelerated down, we feel this as “weight”.
Can we convert between mass and weight?
Strictly speaking – no. Why not?
The same mass will have different weights, when placed on different planets.
Can we convert between mass and weight, assuming that the object is here on Earth?
Oh, that’s different. Sure – for that one planet alone, we can convert between mass and weight. Here’s a conversion that’s valid only on Earth.

1 kg × g = 9.8 N (more exact)
1 kg × g ≅ 10N (approximation)
So these approximate conversions are useful.

1 kg is about 10 N

1/10 kg is about 1 N

100 kg is about a kN
Isolating variables
A literal equation is an equation where variables represent known values.
Variables may represent things like distance, time, velocity, interest, slope, etc.
The slope formula is a literal equation.
Y = MX + B
The distance that an object falls, in t seconds, is a literal equation.
So you’re working on a problem, and you identified the correct formula.
What do we when the variable that you need is not by itself?
In the above equation, how do we get t by itself?
By isolating the variable. In this tutorial, you’ll learn how to do this.
how to isolate a variable.
Learning Standards: Common Core Math
 Common Core Math
 CCSS.MATH.CONTENT.7.EE.B.4 Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
 CCSS.MATH.CONTENT.8.EE.C.7 Solve linear equations in one variable
 CCSS.MATH.CONTENT.HSA.SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. (including isolating a variable)
 CCSS.MATH.CONTENT.HSA.CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.
 http://www.corestandards.org/Math/
Origin of magnetism
Where does magnetism come from?
I’ve heard that special relativity makes the concept of magnetic fields irrelevant, replacing them with relativistic effects between charges moving in different velocity frames. Is this true? If so, how does this work?
Luboš Motl, a Czech theoretical physicist, replies:
Special relativity makes the existence of magnetic fields an inevitable consequence of the existence of electric fields. In the inertial system B moving relatively to the inertial system A, purely electric fields from A will look like a combination of electric and magnetic fields in B. According to relativity, both frames are equally fit to describe the phenomena and obey the same laws.
So special relativity removes the independence of the concepts (independence of assumptions about the existence) of electricity and magnetism. If one of the two fields exists, the other field exists, too. They may be unified into an antisymmetric tensor, FμνFμν.
However, what special relativity doesn’t do is question the independence of values of the electric fields and magnetic fields. At each point of spacetime, there are 3 independent components of the electric field E⃗ E→ and three independent components of the magnetic field B⃗ B→: six independent components in total. That’s true for relativistic electrodynamics much like the “prerelativistic electrodynamics” because it is really the same theory!
Magnets are different objects than electrically charged objects. It was true before relativity and it’s true with relativity, too.
It may be useful to notice that the situation of the electric and magnetic fields (and phenomena) is pretty much symmetrical. Special relativity doesn’t really urge us to consider magnetic fields to be “less fundamental”. Quite on the contrary, its Lorentz symmetry means that the electric and magnetic fields (and phenomena) are equally fundamental. That doesn’t mean that we can’t consider various formalisms and approximations that view magnetic fields – or all electromagnetic fields – as derived concepts, e.g. mere consequences of the motion of charged objects in spacetime. But such formalisms are not forced upon us by relativity.
***
Terry Bollinger , an American computer scientist who works at the MITRE Corporation, replies:
Although the relationship between special relativity and magnetic fields is often stated as making magnetic fields irrelevant, this is not quite the correct way to say it.
What actually disappears is the need for magnetic attractions and repulsions. That’s because with the proper choice of motion frames a magnetic force can always be explained as a type of electrostatic attraction or repulsion made possible by relativistic effects.
The part that too often is overlooked or misunderstood is that these changes in the interpretation of forces does not eliminate the magnetic fields themselves. One simple way to explain why this must be true is that if it was not, a compass would give different readings depending on which frame you observed it from. So to maintain selfconsistency across frames, magnetic fields must remain in place, even when they no longer play a role in the main attractive or repulsive forces between bodies.
One of the best available descriptions of how special relativity transforms the role of magnetic fields can be found in the Feynman Lectures on Physics. In Volume II, Chapter 13, Section 136, The relativity of magnetic and electric fields, Feynman describes a nicely simplified example of a wire that has internal electrons moving at velocity v through the wire, and an external electron that also moves at v nearby and parallel to the wire.
Feynman points out that in classical electrodynamics, the electrons moving within the wire and the external electron both generate magnetic fields that cause them to attract. Thus from the view of human observers watching the wire, the forces that attract the external electron towards the wire are entirely magnetic.
However, since the external and internal electrons move in the same direction at the same velocity v, special relativity says that an observer could “ride along” and see both the external and internal electrons as being at rest.
Since charges must be in motion to generate magnetic fields, there can in this case be no magnetic fields associated with the external electron or the internal electrons.
But to keep reality selfconsistent, the electron must nonetheless still be attracted towards the wire and move towards it! How is this possible?
This is where special relativity plays a neat parlor trick on us.
The first part of the trick is to realize that there is one other player in all of this:
The wire, which is now moving backwards at a velocity of v relative to the motionless frame of the electrons.
The second part of the trick is to realize that the wire is positively charged, since it is missing all of those electrons that now look like they are sitting still.
That means that the moving wire creates an electric current composed of positive charges moving in the v direction.
The third and niftiest part of the trick is where special relativity kicks in.
Recall than in special relativity, when objects move uniformly they undergo a contraction in length along the direction of motion called the Lorentz contraction.
I should emphasize that Lorentz contraction is not some kind of abstract or imaginary effect. It is just as real as the compression you get by squeezing something in a vice grip, even if it is gentler on the object itself.
Now think about that for a moment:
If the object is also charged at some average number of positive charges per centimeter, what happens if you squash the charged object so that it occupies less space along its long length?
Well, just what you think: The positive charges along its length will also be compressed, resulting in a higher density of positive charges per centimeter of wire.
The electrons are not moving from their own perspective, however, so their density within the wire will not be compressed. When it comes to cancelling out charge, this is a problem! The electrons within the wire can no longer fully cancel out the higher density of positive charges of the relativistically compressed wire, leaving the wire with a net positive charge.
The final step in the parlor trick is that since the external electron has a negative charge, it is now attracted electrostatically to the wire and its net positive charge.
So even though the magnetic fields generated by the electrons have disappeared, a new attraction has appeared to take its place!
Now you can go through all of the details of the math and figure out the magnitude of this new electrostatic attraction.
However, this is one of those cases where you can take a conceptual shortcut by realizing that since reality must remain selfconsistent – no matter what frame you view if from – the magnitude of this new electrostatic attraction must equal the magnetic attraction as seen earlier from the frame of a motionless wire.
(If you do get different answers, you need to look over your work!)
But what about the other point I made earlier, the one about the magnetic field not disappearing? Didn’t the original magnetic field disappear as soon as one takes the frame view of the electrons?
Well, sure. But don’t forget: Even though the electrons are no longer moving, the positively charged wire is moving and will generate its own magnetic field. Furthermore, since the wire contains the same number of positive charges as electrons in the current, all moving in the opposite (v) direction, the resulting magnetic field will look very much like the field originally generated by the electrons.
So, just as the method of attraction switches from pure magnetic to pure electrostatic as one moves from the wire frame to the moving electron frame, the cause of the magnetic field also switches from pure electron generated to pure positivewire generated. Between these two extremes are other frames in which both attraction and the source of the magnetic field become linear mixes of the two extreme cases.
Feynman briefly mentions the magnetic field generated by the moving positive wire, but focuses his discussion mostly on the disappearance of the electrongenerated magnetic fields. That’s a bit unfortunate, since it can leave a casual reader with the incorrect impression that the magnetic fieldas a whole disappears.
It does not, since that would violate selfconsistency by making a compass (e.g., the magnetic dipole of that external electron) behave differently depending on the frame from which you observe it.
The preservation of the magnetic field as the set of particles generating it changes from frame to frame is in many ways just as remarkable as the change in the nature of the attractive or repulsive forces between objects, and is worth noting more conspicuously.
Finally, all of these examples show that the electromagnetic field really is a single field, one whose overt manifestations can change dramatically depending on the frame from which they are viewed. The effects of such fields, however, are not up for grabs. Those must remain invariant even as the apparent mechanisms change and morph from one form (or one set of particles) to another.
Does special relativity make magnetic fields irrelevant? Physics StackExchange
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Also see
The Feynman Lectures. 13–6 Magnetostatics. The relativity of magnetic and electric fields