Matter in Space and Time

Section 1: Uses high school math, geometry, algebra, and mensuration leading to kinematics, then to physics of mechanics, dynamics, and in some cases, basics of calculus. You have studied these topics. Browse through some cases!

1.00  Matter in Space and Time

Physics and engineer­ing ad­dress the physical world with the goal of understanding and possibly manipulating reality. However, studies and learning about the real world are restricted, generally, to intense study of books. In effect, early learning about physical reality happens via books - a highly vicarious relation to reality, as it were.

Prove: (A - B) 2 = A 2 - 2AB + B 2

The Greeks understood the totality of anything
to equal the sum of its parts.

An easy way to precede is to begin by with a slightly easier task regarding ( A + B )². The common result was first obtained by Euclid around 300 B.C.

1.01 Basic Methodology

The goal of Engineering Thermodynamics is to "explain" events of physical reality and to predict the future. Thermodynamics is the study of systems and how systems change with energy interactions. Some knowledge groups supporting engineering thermodynamics are physics, mathematics, vectors, calculus, mechanics, language, and other engineering types.

The scope of imformation involved is vast. Its developmenmt has followed a myriad of paths. Also there is no path to follow, to travel retrospectively, to get at the roots or initiation of any topic. Our studies are very much "in the middle." However a "logic of approach" has evolved.

Theorem of Pythagoras

The Pythagorean Theorem, a statement of an algebraic truth regarding the sides and hypotenuse of right triangles, can be proved by use of geometry and the general truth:

"The whole equals the sum of its parts."

Eratosthenes Measures Earth

Knowing geometry and having observed eclipses of the moon, Eratosthanes came to believed Earth was a sphere. To measure its circumference, he devised a plan to used the rays of light form the sun. He assumed all rays from the sun arrived at Earth parallel, each to all others.

Drilling Rig Visibility

drilling rig visibility

The image shows an oil rig at sea which rises 500 feet above sea level and is located 50 "surface" miles from shore. Can the rig be seen from a restaurant atop an 800 foot hotel at the shoreline?

Prove: "The rig can be seen," or tell why it cannot be seen.

To get at it, lets assume some important lady at dinner in the resturant sees the gas flare at the rig and screams, "EGad! Pollution!" By geometry we calculate the distance then compare it to the stated 50 "surface" miles to sea.

Prove: (-1) x (-1) = 1

Most of us use this fact without thinking about it. To prove the formula, we use the results of a previous proof.

1.02 Position: the First Vector

"Where is something" is a question persons have asked over all of time. The answer tells the "position of that something" in a mutually understood space. The "something" might be your car keys or a location such as a freeway exit or the rest room, etc. For a person to answer, "Where is it?," requires both familiarity of both persons with the surrounding space. The space "it is in" must be mutually known.

Pharaoh's Engineers

The Great Pyramid of Egypt was constructed to precise proportions. A hypothesis is that the pyramid was constructed to fit inside an imaginary hemisphere with each of its corners and its peak touching the hemisphere. Suppose the hypothesis were true. Draw a sketch of a pyramid and calculate the angle each face makes with the horizontal plane of the desert.

Vectors Derive Trig Laws

A typical planar triangle is shown. The triangle vertices, lengths of sides, and angles are identified by notation.

We choose to define a vector space to be the plane of the triangle. We name the two-dimensional space 0XY. We place the origin of the space at the A0B vertex of the triangle and we align its X-axis along the triangle side, 0A...


Crank, Rod and Piston

"Power trains" are the mechanical parts by which the power of combustion is transformed into power of a rotating shaft. A sketch of a simple crank-rod-piston arrangement is shown. Engine designers must know the position of the piston face for every position of the crank. The mathematical tool, vectors, makes this task logical.

Ladder-Boom Rescue

Imagine a retirement home is ablaze and flames have backed two elderly women to the opening of a rear, third story window. A ladder truck has arrived in a back alley but a building obstructs its reach.

To place the ladder at the window, it will be extended nearly vertical to the correct, laser determined, length L, then tipped downward to the angle θ. Lasers will obtain the dimensions (given in the sketch) and a computer will solve the geometry. To provide a test case, use the information of the sketch.

Quick Return

The mechanism O-A-B has two links and ground. The input, Bar-OA, oscillates through small angles,­ α ± Δα. Bar-AB, slot-connected to Bar-OA at A, has a changing, active length, X(t). Figure 1. is the basic configuration with notation. Write a general "specification" for all positions of this planar mechanism. Use a math/physics approach that leads to a solution statement readily expressible in computer code.

High Wire Apparatus

After performing with ponies, the dogs of a traveling show do feats on wire strung on a cubical steel frame. The frame measures ten meters on each edge. Once it is erected cables are strung for specific dogs or tricks. Since the feats of the dogs must move smoothly and promptly, the rigger must have a good knowledge of cable lengths and places of connection. In this example we apply vectors to that task.

1.03 Basic Terms and Tools

In the careful progress of science, account is taken regarding not only what is known but also of what generalizations arrive. A vocabulary accrues. Here are some terms of use.

1.04 Models of Reality

The behaviors of masses of real systems are complex. It is virtually always necessary to simplify the system mass. Also it is expedient to engage the simplest calculations; the numbers gotten, though wrong, aid greatly in understanding and how to upgrade the analysis. Very many models of matter are used in analysis of physical reality.

1.05 Motion in Space

It is worthwhile to assume there is some position, some place in the space of the universe that is not moving. A location that stays precisely where it is. No one can prove there is no such location. We assume there is.

1.06 Velocity: the First Derivative

Consider the position of a BODY that moves in 0XYZ - space. At a time (no time specifically but a time, t = t*) an implicit mathematical statement of that position is P(t*) (below left).

This the implicit position of a BODY at a time, t*, in 
space described by the coordinates, 0XYZ.
This the new (implicitly stated) position of the same 
BODY a short time later (at the time, t* + Δt)
in the space described by coordinates, 0XYZ.

Our interest is motion of the BODY until a time later, the inspecific time, t = t* + Δt where the notation Δt is called "an increment in time." By symbol, the position of the BODY at the later time is written (above right).

1.07 Mass Equation: BODY

Although Newton studied physical reality, he studied "simplified" to the vantage of a model. The system of Newton's studies was a constant physical mass, the BODY. It is unlikely that Newton expressed the constancy of the mass as a first order differential equation. Here we do just that. Not to be fancy but to set a sound beginning for the property equations that will be developed. Therefore we write the mass equation of a BODY with in its equivalent mathematical forms.

1.08 Momentum: BODY

In 1687 Newton published two axiomatic statements and three Laws of Motion. His first axiom identified "quantity of matter" as "what he was talking about. Today Newton's "quantity of matter" is known as a mass. Newton (and others) used vectors and calculus to define position and velocity. Newton's Second Axiom defines motion of a BODY:

Axiom II: The quantity of motion is the measure of the same, arising from its velocity and quantity of matter, (mass) conjointly.

1.09 Derivative of Momentum: BODY

To explain "motion of a BODY," Newton invented the derivative of calculus and with that a special spatial characteristic of a BODY; its velocity. With those ideas as basis, Newton postulated a second axiom.

Axiom II:  Quantity of Motion of a BODY is defined to be the measure of the same, arising from its velocity and (quantity of matter) mass conjointly.

1.10 About  f = ma

In 1687 Newton published his famous Three Laws of Motion. Every two or so years thereafter, some scholar, physicist or engineer has published an article or text chapter in which "What Newton Meant," is explained. That debate continues. A recent, representative writing (which I paraphrase here) is Richard Fitzpatrick's Newtonian Dynamics.

In presenting Newton's 2'nd law to students, HS physics avoids the vectors and calculus Newton invented and used. Most physics texts reduce Newton's understandings of motion to an algebraic equation. Three equational notations used in physics texts are:

HS text forms of Newton's Second Law of Motion.

1.11 Uniform Motion

Two characteristics of a BODY (in an 0XZ space) are its position, P, and its velocity, V. Two properties of the BODY are its mass, m, and its mass-velocity product, its momentum, mV. Newton used calculus to develop a single description of motion of a BODY in space. Today his First and Second Law are aptly expressed by a single differential equation.

1.12 Constant Momentum Motion

To say the momentum of a BODY is constant is to say "at the initial instant of observation" the BODY had a certain velocity which it continued to have thereafter. Since the mass of a BODY is constant, its momentum, mV, is constant provided its velocity is constant. For constant, non-zero velocity, the event of the BODY is said to be "uniform motion." The special (sometimes called "trivial") case is the BODY with initial velocity (and thereafter) equal to zero, The BODY is said to be at "rest."

Valentino's Wake

When Rudolph Valentino died (1936) his body was displayed at his wake in an open casket. One newspaper report asserted:

"Today 9,000 persons per hour passed
Valentino's casket in solemn tribute."

Is it possible that 9,000 people walked past his casket in an hour?

Dog Greets Owners

Problems in HS physics texts some­times omit detail then request an "answer" when only an "approximated" answer is obtainable. For example!

A dog runs back and forth between its two owners, who are walking toward one another. Initially the owners are 10 meters apart, the dog runs at 3 m/s and the owners each walk at 1.3 m/s. How far will the dog have traveled when the owners meet?"

Yacht and Sea Buoy

A yacht sets sail eastward from a dock at a speed of one mile per hour. A sea buoy is anchored two miles northeast of the dock. Our event starts at t = 0+ with the yacht (position and velocity) and the sea buoy (position) as indicated in the scenario sketch.

When will the yacht pass closest to the Sea Buoy?


This sketch is a physical scenario. It depicts what will happen.
Looking at it closely, we anticipate the lift-rate of the scissor-jack 
will vary during any lift with constant input rotation.

A scissor-type car jack is shown. Member BC is threaded. When it is rotated by the hand-held driver, the distance |BC| shortens which in turn causes the top of the jack (with elevation, h(t)) to move upward. The lead of the threaded bar BC is 0.1 cm/rev, meaning one rotation of the screw shortens the distance |BC| by 0.1 cm. For the position shown,

Calculate the upward velocity of the jack, point D.

Train Passes Two Boys

Two boys, walking beside railroad tracks heard a train approaching from behind. The older boy knew city train speeds were limited to 30 mph and that he and his buddy walked about 3 feet per second. When the nose of the engine was abreast of them, the smaller boy began to count. The count, the instant the caboose passed, was, "... 34 seconds." A moment later, the older boy said, "... only about 1400 feet long."

Civil War Memorial

Many statues have been erected in honor of the soldiers who died in the Civil War. A typical monument (this one in Mulberry, Tennessee) is shown. Suppose you were to observe the original and to notice that it was at rest.

Model the statue as a BODY. What does Newton's Second Law of Motion tell us?

1.13 Steps to Integrate

Of calculus, at this level, we need know only the definition and meaning of Newton's derivative. Physical laws for thermodynamics are expressed as first order ordinary differential equations. Presented below is a simple, time-dependent first order differential equation with its initial condition. We solve the differential equation then we show (in reverse) the differential equation that belongs to our solution. This is the differential equation we solved in the first place, of course.

1.14 Measurement of μMEarth

Newton postulated that all masses experience forces as a consequence of the Earth mass. He called this phenomenon, gravity. Gravity forces which act on all parts of mass of a system are directed toward Earth. The magnitude of the Earth gravity force was postulated by the equation. The challenge was to make this force measurable or quantitative.

1.15 Weight and Weighing

The phenomenon of mutual attraction between masses and the very great mass of Earth is called "gravity." Newton recognized that a gravity force acted on every system (BODY) as a consequence of its mass. The magnitude of the gravity force equals the "weight" of the BODY. Hence "weighing" or to weigh a BODY is to measure the magnitude of the attractive force of Earth.

Hand Supports a Mass

The "standard" acceleration of gravity (at sea level and 45° latitude) is 9.80665 m/s². What force is required to hold a mass of 2 kilograms at rest in this gravitational field? What mass can a force of one Newton support?

Lunar "Carry-Off" Luggage

Suppose on Earth an astronaut, using his maximum "lifting force," lifts 30 kilograms. Since unloading on the moon will be easier, let's use a forklift to load on Earth. What should be the mass such that the astronaut can apply the same magnitude of force to unload on the moon?

Scenario shows person holding pota-
toes. To the right is a FBD in which 
the ambient density of air is assumed 
equal to zero.

Ten Pounds of Potatoes

When English Engineering units were specified (~1824) mass was set to be a dimension and defined as a certain, specific quantity of matter. The "founding mass" was a cube of metal kept in a vault in France with replicas for distribution. In those times "force" was an idea more subtle than mass. One immediate idea of force was the "effort" required by someone to hold or support some mass in his hand. Some clever lads supposed force to be a dimension as is mass. Going further, they gave quantification to force. They defined a "unit force" (one pound force) to equal in magnitude, "that force" required to support a unit mass (one pound mass) at sea level.

1.16 Vectors: BODY

Typical vectors for a BODY (as system) are its spatial properties: position and velocity (also acceleration). The gravity force and sum of surface (or external) forces, ΣF, are vectors. The figure shows a BODY in 0XZ-space with a notation for its vectors.

1.17 Vector Notations: BODY:

Some say position and velocity are system "characteristics," not properties. It is helpful to use simple but effective vector notations with beginning mechanics and thermodynamic analyses. The table presents some representations of position and velocity (of a BODY or a Point).

1.18 Value, Slope and Curvature

Science uses mathe­matics to describe events of physical reality in time. Here is discussed "function of time." Newton viewed reality and its mathematical description to be the same in the limit of very change, short instances in time. At an instance, say at t = t*, physical reality described by a mathematical function has a value, a change of that value (function slope) and a curvature. Value, slope and curvature, that mathematics, explains, physical behaviors at their basis.

1.19 Differentiation: Time Dependent Integral

This simple proof leads to the idea “dummy variable”.
Also the “Mean Value Theorem is used."  This is important 
calculus for physics.

Time is relevent to every physical analysis. The basic characteristics of of motion are position, velocity and acceleration. It is inportant, given any of the three, to integrate ofr differentialt (as the case might be) to obtain teither of the other two. Of course the relations are sound. But the calculus can get a little sticky. This partial review of basic, definitional, calculus.

Blue Ocean Towing

Our company has contracted to protect an off-shore oil rigs. In the Davis Strait, a massive ice slab (~ 800x106kg) has cleaved from the ice-shelf and is drifting with the sea current directly toward the rig.

The map (right) shows the initial location of the the slab relative to the the production rig. We plan that our largest tug, (pulling constantly at 90° to the current), will drag the slab off-course such that it will pass, abreast of the oil rig, at a distance no closer than 4000 meters.

Calculate the towing force the tug must sustain to accomplish the task.

1.18 Gravity at Altitudes

The attention of physics and engineering texts to the purported "significance" of the variation of Earth's gravity with altitude is exaggerated. Those text problems ignore the inconsequentiality the effect has for the overwhelming majority of events of engineering systems.

Projectile Arcs Intersect

A body is projected from a point O on horizontal ground with speed of 70 m/s. It passes through a point P, which is 45 m above the ground and 50 m horizontally from O. Calculate the tangents of the two possible angles of elevation of the projected body. Find also the gradients at P of the two paths.

1.21 Events in Time

Time, perhaps the most grand of all abstracts, attends all thermodynamic events (processes) but it is not a property of any system. Time is easily made quantitative on a relative scale. It is helpful to think that time has a timekeeper with a "clock." By "starting a clock" (initiating its first sequence of periodic events) then counting those repeating, we use clocks to establish what happened first, second, thereafter and so on. Time does not participate in events, it is an observer. But "time" as science uses the idea, is not a casual observer. The "time" of science is a "batch" perspective.

Galileo's Experiments

When Galileo attempted to measure the free fall acceleration of a BODY in Earth's gravity he found its effect in vertical fall too fast to measure. To diminish the effect, to slow the event, (so as to be measurable), Galileo constructed a very smooth, rigid, inclined plane.

Given the measurements of one event, calculate the value of Earth's gravity, go.

Acceleration - Initial Condition

A problem statement, regarding dynamics of a BODY, contains specification of the acceleration of the BODY at the instant of initiation of its event as a "given." Thus the author implies, by some physical means or manner, "at the commencing instant of an event" the acceleration of the BODY is determined. Below is a discussion of how, physically, that "initial condition" might be known.

1.22 Omitted Forces

In solving problems, drawing sketches and in reading texts, one must be on the guard against carelessness. One critically important error to avoid is omission of forces. It is better to use the notation ΣF rather than F. The summation sign is a reminder to include ALL forces relevant to the physical situation. Students and (authors) who use simply F or f for forces tend to omit relevant forces more often than those who use the notation: ΣF.

Parachutist's Drag Force

At the precise instance of leap, the acceleration is Earthward at 9.8m/s². A few moments later, in free fall, the acceleration of a parachutist becomes 6.8 m/s². The mass of the person is 90 kilograms. Calculate the frictional drag force of the surrounding air at that instance of the fall?

GOD Lifted Earth I

This approach takes the Woman and
Earth together as the system.

In Massachusetts (1710) an accused witch was hanged in the public square. At the moment of her death, when that rope snapped taut, an experience occurred to her family and loved ones, they believed an earthquake had happened. At night, in full darkness, friends cut her down and a rumor got started. "When she died, God lifted Earth to receive her," whispered one bystander to the hanging. Within a day minds changed; the entire village came to believed she was innocent.

GOD Lifted Earth II

Often, to understand an event, it is convenient to construct a simpler model to solve. The sketch to the right is a 'physical scenario" similar to the previous example, "GOD Lifted Earth I." Our sketch shows two spheres of equal mass held in a state of constrained equilibrium by a rigid frame and cord. We assume these are the only masses and that the surface gravity of each is represented by go. Think of the cord as a catalyst. When it is cut, an event will occur.

Point Blank

Warships abreast, fewer than 300 yards apart, were in the firing position Lord Nelson's crews called "point-blank." Suppose Ship A fires a cannon, aimed perfectly horizontal, at Ship B. The shot mass 24 pounds. Assume its muzzle speed constant over its flight at 500 ft/s. What does Newton's Second Law tell us about "Point Blank?"


Virginia Shot Tower


From years before the Revolutionary War and through the Civil War over a million spherical lead shot for muskets were manufactured at the shot tower in Wytheville, Virginia. The shot sold in twenty-five pound bags. Of the lead shot produced, 10% failed to qualify and were remelted.

What least mass of lead was melted per bag of shot produced?
Approximately how many shot would a bag contain?
Calculate the speed attained by a shot that falls for three seconds.

1.23 Momentum Equation: BODY

Newton's axiomatic approach and his Laws of Motion, as taught in HS physics, might be said to be the beginning of the scientific method. However, to proceed to to a full usage of Newton's ideas it is helpful to cast his law, "f = mA," in new perspective.

Today, some 300 years later, Newton's "Second Law" (in an inertial reference) is what is used. And it is perfectly adequate except for its name. "Second Law" makes one wonder, "What's the "First Law?" A better vantage would be helpful. The new vantage proposed here is:

1.24 Momentum Components: BODY

Newton's Laws of Motion are vector differential equations. The most common coordinate system used for their expression is Cartesian coordinates (OXYZ) with origin at the surface of spherical Earth and the positive Z-axis vertical upward. This space has three two-dimensional subspaces, OXY (with Z-symmetry), OXZ (with Y-symmetry) and OYZ (with X-symmetry). For the purposes of study the perspectives "OXZ" or "OYZ" are alike: gravity acts in the Z-coordinate. Interestingly, when one views physical reality and occurrence choosing the horizontal, OXY, two-dimensional plane, gravity is excluded.

Geostationary Orbits

Modern rockets routinely place communications satellites in Earth orbit such that they maintain a constant position above and relative to Earth. Such orbits (which are possible only in the plane of the equator) are accomplished when the rocket "parks" the satellite at the proper altitude and with the proper velocity. The satellite is parked ina Geostationary orbit. How high up is that?

1.25 Vector Basis: Circular Motion

Motion of a BODY at constant speed constrained by forces to follow a circular path is kinematically described by the radius of the circle and vectors that depict its position, P(t), velocity, V(t), and acceleration, A(t).


A caution of physics texts about usage of Newton's 2nd Law is: "Be sure to use a non-acccelerating, that is inertial coordinate reference." However, after that state­ment, through numerous examples next to nothing further is said or done regarding inertial coordinate reference selection. The examples, being in an Earth environment, does text disregard of reference imply all places on Earth are inertial?

Upon further reading one encounters a second "caution." If coordinates "not inertial (by error or choice)" are used, special "fictitious" or "pseudo" forces might be needed. Centrifugal force is the simplest case.

Polar Versus Equatorial Weight

Suppose we have a certified mass of 1,000 pounds. Demonstrate analytically that the mass will weigh 1,000 pounds force at the North Pole but only 996 pounds force at the equator? (Ignore the buoyant effect of atmospheric air).

Mass and Sled

The sketch is from a physics text. According to the drawing, there is one horizontal force that acts with magnitude, F, directed to the right. The implication is acceleration of masses m and M occurs "at the very instance" of observation.

Calculate: The greatest force such that NO sliding of the mass on the sled will occur.

Kinematics of Bar-AB

Typical text-book "given-information" sketch.

"Bar-AB and Ground" constitute a kinematic mechanism. Connections at A and B (not shown) maintain sliding contact with the ground. At the instant shown, End-B has a velocity of 0.5m/s and an acceleration of 0.3m/s² (in the directions shown). Calculate the angular acceleration of the Bar-AB and the acceleration of End-A.

Lift-Off Acceleration


Motion char­acteristics of a rocket during its lift-off are observed by a tracking laser housed in a blast-proof bunker at a distance from the launch pad. The figure shows the initial situation.

1.23 Method, System and Numbers

The grandest abstract of thermodynamics is the universe which is defined to be all that exists in a physical sense. Everything that happens, happens inside the universe. Being infinitely vast, the universe extends well beyond the range or the localized volume of our engineering systems and events. Physical events of importance to human life occur in vanishingly small spaces in the universe. The impossibility of dealing with the entirety of the universe mandates a system approach.

1.25 Newton's Analytic Method

To write a summary of Newton's approach and to include the extensions of his work made to this day is a lofty goal of writing. Stated otherwise, "What Newton said and where we are as a consequence" is not a light topic. But we give it a try.