A previous post on Work and Energy concluded with the following statements.

Even if a force is not conservative, such as friction, the work done by that force represents a transfer of energy from one form to another. In the case of friction, large scale motion is degraded into random motions of atoms and molecules, which constitute heat. And heat is a form of energy. In fact, there are many forms of energy.

So far, physicists cannot find any sources or sinks of energy in the universe. The transfer of energy from one form to another may look like a source of energy, or a sink for energy; but if you look hard enough you can always trace an unbroken chain of energy exchange in a zero sum fashion, the total of which does not change in this universe.

In fact, there is a deep theoretical reason for the conservation laws, such as that of energy and momentum, which comes from Noether's Theorems: wherever there is a symmetry in the laws of physics there is an implied conserved quantity. In the case of energy conservation, it arises from the laws of physics being invariant in time. The laws of physics being the same if the whole laboratory is translated in any direction and any amount implies that momentum is conserved. No matter how the laboratory coordinate axes are oriented, the laws of physics remain the same. In that case, angular momentum becomes the associated, conserved quantity. So there is a deep connection between symmetries and conserved quantities, between dynamically conserved quantities and time and space.

Because energy is conserved, the idea of elastic and inelastic collisions really applies just on the large scale, where ordinary objects are seen to lose their motions in a short time: tops stop spinning; grandfather clocks wind down and have to be rewound to keep going; bouncing balls come to rest. So in some sense, all collisions among everyday sized objects are a certain fraction elastic and the remaining fraction, inelastic; but when the degradation of the motion of macroscopic objects is examined microscopically, all processes are seen as elastic. The degradation of motion on a large scale is seen as the transfer without loss of energy on the large scale to the smaller constituents of the large objects. In this sense, all collisions are elastic. Sometimes to be able to see this we must use all the physics that we know.

One modern subtly is that Einstein's equivalence of mass and energy is at work in high energy collisions of particles in a modern particle accelerator. Two protons moving at energies equal to a plethora of other particle rest masses in undergoing head-on collisions can produce a jet-spray of a bunch other high energy particles not originally there, but created by the transfer of the kinetic energy in collision into the moving masses.

There are several parts to the modern view of energy and momentum. Energy is substance. A particle of matter is made up of energy that is held together in a certain form. This form is not just in time and space, but involves configurations in terms of additional abstract quantities.

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At one time in the history, classical physicists were developing the concept of energy, momentum, and their conservation (see a previous post). Most of the observed mechanical motion in laboratories decayed fairly rapidly because of friction. Using oil and good bearings, mechanical designers could slow the loss of energy. Although unrealizable in practice, with mathematics, these early researchers could model motion without friction. In some cases, the motion could be broken down into two components: kinetic and potential energy.

A place where the concept of frictionless motion proved practical was in the case of the motion of planets and stars. Armed with Newton's laws of motion and that of gravity, scientists developed the discipline of astrophysics, which has proved itself accurate enough to guide our rockets to various places in the solar system. and even beyond. This branch of physics describes the motion of the planets very well. It fails only because of the limitations of Newton's Laws of motion, which are now replaced by Einstein's General Theory of Relativity. General Relativity is generally over-kill in describing planetary motions, except in the case of Mercury, the orbit of which is successfully described by Einstein's General Relativity, but not Newton's theory. Nevertheless, astrophysicists using classical mechanics have been able to devise orbital strategies that further accelerate probes by using fly-bys around various other celestial bodies to boost their orbits though a sling shot effect.

An interest in collisions developed early in the history of physics and continues into present day physics. Today we use high energy accelerators to maneuver elementary particles moving a close to the speed of light into head-on collisions to test the predictions of elementary particle theories. In this post, we study two types of collisions: elastic and inelastic. Although in themselves they are more of historical and pedagogical value, the concept has a philosophical value that I think can be used to clear up various mysteries surrounding weird quantum behavior.

Consider the conditions depicted in Fig. 1, which illustrates conditions existing just before mass 1 (on the left) impacts another stationary mass 2 (on the right). Physicists find that a transformation of coordinates often simplifies and clarifies a problem. In this case consider a simple Galilean transformation to a reference frame moving with the center of mass (CM) of m1 and m2, which is pictured in the upper part of Fig. 1. That transforms the reference frame into one, in which the CM is at rest (lower part of Fig.1) and both masses are initially in motion. See the post, Grokking Galileo's Physics I, for motivational clarification.

In the CM coordinates (lower part of Fig.1), the CM itself is at rest and both the masses are initially approaching each other along a horizontal axis to a point where they will collide.

Below, the two types of collisions are discussed referencing the CM frame of reference (Fig. 2) for: 1. an inelastic and 2. an elastic collision. But first notice that the momenta of the two masses in CM coordinates add up to zero, the entire momentum of the system in laboratory coordinates having been invested in the total of the two masses moving with the center of mass.

In the inelastic collision (1) depicted in Fig.2, mass 1 has penetrated and stuck to mass 3. The combination of the two becomes a stationary total mass in the CM frame of reference.

The elastic collision happens when the masses are very elastic and the two masses bounce off each other completely reversing the momentum of each mass. The total momentum of the masses goes from zero to zero. Momentum is conserved in the collision, but note that the elastic collision results in the parts having more energy than the inelastic collision.

In the inelastic case, the kinetic energy of the two particles has disappeared, having been used up in mass 1's penetration of mass 2. The result is that the total mass is at rest relative to the CM. Whereas, after the elastic collision, there is motion left over in the masses moving relative to the CM.

This about the full extent of what we can discuss without the use of mathematics. In the mathematical discussion that follows, it is shown that both types of collisions conserve momentum, but in an elastic collision total energy is conserved, but not in an inelastic collision.

This part of mathematical discussion analyzes the motion before collision, where mass 1 is in motion approaching mass 2, which is at rest (Fig. 1). No external forces act on the two mass system, so that its center of mass (CM) moves at a constant velocity. By using a Galilean transformation from this frame of reference (the laboratory coordinates) to the CM frame of reference, it is shown what could have been grokked that the total kinetic energy consists in the motion of the CM of the two particles plus that of the two masses relative to the CM.

We will adopt the following notation to keep the discussion compact

momentum,

p=mv (1)

kinetic energy,

KE=½ mv^{2 }. (2)

The above are generic notation that are specified to the objects in question through subscripts. Further, the notation is specialized to motion along a single axis. To use a full 3D notation would be an over complication of the discussion. An example of the use of the above notation to specify the pre-collision state of the system is as follows:

p_{1} =m_{1} v_{1 } (1.1)

KE_{1}=½ m_{1} v_{1}^{2} . *(*2.1)

Since only mass 1 is moving initially, (1.1) and (2.1) describe the state of motion of the two mass system. In what follows, bold, capitalized symbols the motion associated with the two particle system,

**P **= **M **v_{CM }, (1.2)

where v_{CM } is the velocity of the center of mass (CM) and

**M** = m_{1}+m_{2}, (3)

is the total mass.

The total kinetic energy associated with the motion of the CM is exactly equal to that of mass 1, which was the only one moving in the laboratory coordinates,

**KE** = KE_{1 } . (4)

The kinetic energy associated with the movement of total mass of the system is also given by the following equation:

**KE** = ½ **M** v_{CM} ^{2}. (4b)

To carry out the Galilean transformation from the laboratory coordinates to those of the CM, we need the CM velocity, which is found by equating (1.1) and (1.2). The result is

v_{CM}=v_{1 }(m_{1}/**M**). (5)

Now apply a Galilean transformation to the two particles to a coordinate system, in which the center of mass is stationary:

v_{1}'= v_{1}-v_{CM } (5.1)

and

v_{2}'= -v_{CM }. (5.2)

It is left for the reader to show that the total momentum in the CM frame is zero,

m_{1}v_{1}' + m_{2}v_{2}' = 0. (6)

Before the collision, however, both masses contribute to the kinetic energy in the CM frame of reference,

**KE'**=½ m_{1} v_{1}' ^{2} + ½ m_{1} v_{2}' ^{2}. (7.1)

Substitute for the transformed velocities using (5.1) and (5.2) and after a bit of algebra, the reader should get the following equation,

**KE'**=½ m_{1}v_{1} ^{2} +½ Mv_{CM} ^{2}- m_{1} v_{1 }v_{CM} (7.2)

Equating momenta in (1.1) and (1.2) reduces (7.2) to the simple one below,

**KE'**=½ m_{1}v_{1} ^{2} -½ Mv_{CM} ^{2}. (7.3)

By rearranging terms in (6.3), we get the simple expression.

**KE**=**KE'**+KE_{CM} , (8)

which indicates that the total kinetic energy of the two masses before collision is decomposed into two parts by the Galilean transformation: (1) the part carried by the two masses in the CM frame, and (2) the part carried by the center of mass motion relative to the laboratory frame.

This part of the discussion concerns what happens after the collision, which an be either elastic or inelastic.

Most of the mathematical part of the discussion is done. The rest is easy. First, we consider the easiest case, where mass 1 penetrates mass 2 and sticks to it (Fig. 2). The two combined masses in the CM system are now at rest, straddling the CM between their own centers of mass. As a result of this inelastic collision, the kinetic energy of the two masses relative to the CM just disappears. The CM relative momenta of the two masses are always equal and opposite; therefore, their momenta add up to zero, no matter the magnitude of those momenta. Meanwhile, the CM cruises along relative to the laboratory frame without change, the kinetic energy (**KEi**) of the two particle system (8) having been reduced to that of the CM motion alone, which we evaluate as follows:

**KE _{i }**= KE

**KEi **= ½ **M **v_{CM} ^{2}.

**KEi **= ½ m_{1} v_{1}^{2 } (m_{1}/**M**).

**KEi **=KE_{1} (m_{1}/**M**).

In an elastic collision, not only does the total momentum of the two particles remain unchanged (6), but also the total kinetic energy remains the same. How can that be? Just look at the two masses in the CM system in which the two masses come crashing together from opposite directions. The momenta of the two are equal in magnitude but opposite in direction. After the collision, each mass has undergone a total reversal in momentum. Now both are flying apart with equal and opposite momenta of the same magnitude that they had before the collision. The effect has been that each velocity has been changed to its negative. Because the kinetic energy of the two masses in the CM system (7.1) depends on the square of the velocities of the particles, a reversal in the sign of both is not going to change their kinetic energy. Therefore, the total kinetic energy of the two particles (8) remains unchanged as result of the collision.

The effects of collisions are demonstrated by using a device called Newton's Cradle. Look at this video to see such a demonstration. Although true elastic collisions happen mostly on a quantum mechanical level, it is possible to approximate them on a macroscopic scale using highly elastic substances. In 1965, a toy called Super Ball came out on the market. It had an amazing bounce. Each bounce on a hard surface was 92% the height of the former. Because the energy of a bouncing ball is the sum of its potential and kinetic energy and its total energy is potential at the top of its bounce, the Super Ball lost only 8% of its total energy with every bounce.

IF you want to see what goes on in a ball during a collision, see the high speed tests of a struck golf ball.

This discussion is continued in the next post, "Work and Energy--Where does the Energy Go?"

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When I was 24 years old (just a kid) I spent about month in Europe. I had been stationed at Aberdeen Proving Ground at the Ballistics Research Laboratories as a scientific programmer on the ORDVAC digital computer. This was part of my military stint after having been drafted away from my work at White Sands Missile Range, where I started my career after gaining a Bachelor of Science degree in physics. We called the base, 'the campus'; and the area, during hot summers, the armpit of the nation.

I am a Westerner, and being east of the Mississippi, I took every opportunity while in the Army to visit the sights back East. My goals on graduating from high school were: to be renaissance guy; to go for a PhD in physics; and to see some interesting parts of the world. My interests beyond science, were classical music, literature, fine arts, and history of the western world, going back to pre-history, such as in cave art. Of course, I went to the Smithsonian in D.C. and visited various galleries, including the Guggenheim. But I also did ordinary stuff, such as eating at a deli, or at the Horn and Hardart, and going to off-Broadway plays. Having done all this, I found out that there was opportunity to do much more than this. I could go to Europe by Military Air Transport (MAT) flights. My ticket was the uniform and orders cut by my company captain that would allow me to go to Europe. The result was that I spent about a month traveling in Europe, sometimes by rail, sometimes by air, and often by automobile (a red Porsche convertible), see post, "From my Journals--A trip to Europe."

During my European trip as a young man, I made a couple of jaunts to Spain, first to Barcelona, then to Madrid. The first trip is described in my post, "On to Barcelona". Someday, I may write a few things about my experiences then in Madrid. Since those early days of my youth, I have visited Spain several times during professional travels, and I have written a number of posts about later experiences n Spain. Looking back over the blog offerings, I see that I have written quite a bit about Spain and related items. If you are interested in reading these entries, just follow the links below:

http://www.ghyzmo.com/the-caracena-name-in-history/

http://www.ghyzmo.com/town-of-caracena/

http://www.ghyzmo.com/views-of-the-town-of-caracena/

http://www.ghyzmo.com/history-of-the-town-of-caracena/

http://www.ghyzmo.com/the-origin-of-the-caracena-family-of-spain/

http://www.ghyzmo.com/the-caracena-family-during-the-time-of-discovery/

http://www.ghyzmo.com/the-sephardim-of-spain/

http://www.ghyzmo.com/Salamanca/

http://www.ghyzmo.com/second-sons-and-the-spanish-empire/ .

My initial interests in travelling in Spain joined naturally with later interests in genealogy. Interests in family history motivated me to study the history of the Spains, Spain, and the Spanish Empire. In subsequent posts, I may discuss more the mechanics of touring in Spain, which would perhaps be of more interest to the average reader.

Recently, my niece Kathy walked the "*Camino de Santaigo*", or Way of Saint James that extends from the French side of Navarre to Santiago de Compostela in Galicia, Spain. This is an ancient pilgrimage route that many Europeans have traveled since the Middle Ages. I wish that I had traveled this way as a young man. The movie starring Martin Sheen called, The Way, is a great way to experience some of the excitement of this pilgrimage. If you have more active interests, go there and travel in Spain yourself. If you want some more ideas on what to do there, look at the following website: https://www.your-rv-lifestyle.com/best-things-to-do-in-spain/ .

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A recent headline on an AccuWeather article reads as follows: "

The quoted article discusses the conditions expected at the surface, which will be miserably hot and sticky back East. DC will live up to the famous saying, "Capital punishment is having to spend the summer in the nation's capital! " Oddly, the mechanism left over from a cool wet spring is partially responsible for this flash heat wave. This mechanism is an overactive jet stream unusually far south for this time of year because of a blocking high pressure area over Alaska.

The above figure illustrates the situation as of the evening of July 18 2019, which corresponds to July 19, 0000 UTC (the old GMT and military Z, see article here). The figure depicts the temperature distribution as color filled areas (red for hot, and blue for cold) and height contours (in tens of meters) of the 500 mb pressure surface. This pressure surface, which is close to the mid depth of the atmospheric mass is often used by meteorologists in analyzing weather situations. This figure was generated by an objective analysis program that numerically analyzes numerous balloon soundings of the atmosphere, in this case, released at 0000 UTC. Note that the height contours in the figure have two interval spacings, 100 meters below 5800 meters, and ten meters above this level to contour the top of the ridge across the South. The course spacing for the lower parts of the surface was chosen to not obscure the pattern with a dense pattern of too many contours.

An advantage of depicting the height field of a pressure surface is that the local density of height contours indicates the strength of the wind there, which generally follows the direction of the contours. In the present situation, the height contours are compressed across the northwestern tier of states indicating a jet streak moving east. Initial conditions from global NWP (**n**umerical **w**eather **p**rediction) models (not shown) indicate that this jet streak is part of a longer jet stream that is arriving from over the Pacific. The jet stream is situated over a sharp front, which separates unusually cold air to the north from unusually hot air to the south.

The situation painted by the figure is that there is a dynamic thermal separation maintained by the dynamics of a jet stream unusually south of its normal position, which is fighting against the normal expansion of a continental ridge northward at this time of year. What we have here is the famous kid's conundrum: "What happens when an irresistible force meets an immovable object?" A subsequent post will continue this discussion by addressing the question, "why is the hot temperature at 500 mb such a big deal for us humans down on the surface of this blue planet?" There is an important topic here that connects with the global climate.

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