1.3 POSITION, VELOCITY AND MOMENTUM

Our next task is to understand (at a beginning level) the system characteristics, position and velocity, and the system property, momentum. To understand mechanics and thermodynamics requires skill with mathematics, some vectors and a small amount (the sweetest part) of calculus. These skills will be used repeatedly throughout this writing.

POSITION

Position, in its very simplest meaning, the simplest answer to the question: "Where is it?" The "it" is aidefinit. It might be a freeway exit, your car keys or the rest room. The answer, the "position of it" can only be known (or told) in relation to prior knowledge - the previously defined mutually understood space of us and it. First, a mutually known vantage of observation such as "the court house" (technically called the origin) is required. Reference directions attached to that point, such as north, east... must be specified. To employ mathematics, a vector basis, "scribed" in a permanent way, aligned with directions at the origin, is needed. The courthouse as origin, the directions and the vector basis constitute what is called a "well-defined vector space."  

Vectors and the Scalar Product:  The figure shows a triangular shape implicitly; implicit meaning with no numerical information included. An origin, orthogonal axes (in the horizontal 0XY plane) are drawn and the triangle vertices are labeled 0, A and B. The triangular figure is "implicit" in that no angles, coordinates of points, etc, are specified. Suppose the shape represents an actual piece of land with small stones (point masses) placed at A and B. The positions of the points and their relation each to the others can be written "implicitly" in terms of directed line segments, i.e., vectors. A common notation is to write vectors as "ordered pairs" of letters.

Implicit Vectors   An implicit vector representation of the triangular area results when "arrow heads" are applied to the previous figure. By adding an arrowhead to one end, of a line segment, one changes the segment into a vector magnitude (length of the segment) and direction. In our vector space, the position of the stone A, can be written as, 0A. The ordered pair of letters, 0A, in a vector sense, mean a directed line segment that initiates at 0, (the origin of the vector space, and extends a distance to terminate at A. Similarly, stone B has a position located by a line that commences at 0, and extends to B, vectorially written as 0B. Vectors starting at places other than 0 follow logically. A vector commencing at A that extends to B would be written AB. And since vectors are additive and span space, it stands to reason that:

0A + AB = 0B

The equation reads left to right to state that the sum of two vectors equals a third vector. To solve or discuss any vector equation is made easier by focusing first on the easier of the vectors to describe. In the above equation, vectors 0A and 0B are easier to discuss. Each vector starts at the origin, 0, and procedes to its location (A or B, respectively) where their vector heads, or arrows are drawn. Vector AB is slightly more complicated; it proceeds from A to B. In this notation vectors are represented as ordered pairs of letters. There is a rule for adding these vectors. Two vectors can be added provided the ending letter of the first is the beginning letter of the second. The vector sum, BH + HS, is valid and equals BS. The idea extends: 0C + CD + DF = 0F.

To this point the vectors we have discussed were implicit meaning clearly stated but not yet quantitative. Vector equations are created implicitly, double-checked then changed to explicit form to be solved. It is easier to learn about physical vectors in focused, physical contexts. And, although space vectors are three dimensional, often at two-dimensional coordinate space will suffice. Learning is not harmed and an economy of vector arithmetic is obtained.

Explicit Vectors -   Next we make our triangular figure more specific by specification of numerical values and insertion of a unit vector basis. For 0XY-space, the pair, I and J are used. We see that 0A and 0B can be written as:

0A = 9 I + 0 J    and     0B = 12 I + 5 J

The above equations show simply the implicit vector OA is explicitly written as 9I. Also the implicit vector, 0B, has an explicit representation in component form as, (12I + 5J). This being a two-dimensional space, a basis and two numbers are needed to specify a vector. Alternately if the vector is not known, two numbers are not known. Often it is convenient to write a vector in a second way, in a "magnitude-direction" form.

The magnitude or length of vector 0B implied by writing the vector bounded on both sides by a vertical bars, as | 0B |. To obtain that number, since the components of 0B are orthogonal, we calculate as follows:

Thus the length of vector OB is "13" units. To determine the direction of vector OB we multiply it by 13/13, then move the "13" of the denominator to the right, beneath each of the component numbers.

Thus, in summary:

A final point is the quotients, 12/13 and 5/13, respectively are the cosine and sine of the angle α which equals 22.6°. 0B can be written using that angle.

So these are some of the representation styles of vectors. With use, the advantages of one style or the other are learned.

Scalar Multiplication of Vectors.  Vectors can be multiplied in a scalar manner or a vector manner. We will want to know about scalar multiplication. That operation proceeds as follows:

This simple operation shows that scalar multiplication of two vectors produces a scalar which equals the magnitude of one vector times the magnitude of the other, times the cosine of the smaller angle between them. Were it the case that the vectors were perpendicular, the cosine of 90° would equal zero, hence the scalar product is zero.

As illustrated above, vector equations relating positions of points in space are easily written. And, once written, they are easily checked. Vector equations, written correctly, express the geometry of any physical situation, any event,in a powerful, orderly manner. The next example illustrates the power of a vector approach.

Crank, Rod and Piston   The mechanical parts of a single-cylinder gasoline engine work together to transform combustion energy dilevered at the piston face to rotary power at the drive shaft. Engine designers must know the position of the piston, 0A, for every position of the crank, 0B. The operating variables are the angles, α and β. The design constants are the variable length, a, and the constant lengths, b, and c. Describe the engine components with vectors in two dimensions. Reduce the vectors to obtain two algebraic equations that relate the angles and lengths of the mechanism.

The solution of the above example shows that the vector approach includes facts of trigonometry, inherently. Vector equations that relate points in space are easily written, easily checked and inherently express the geometry of any physical situation in a powerful, orderly manner. The next example is about the development of laser guided controls for the boom of a Rescue Truck. A computer needs a vector algorithm and a sample set of data for calibration of the control system. The following example obtains "answers" for one scenario of emergency.

Ladder- Boom Rescue Truck:   This example addresses the calculations necessary for development of a control system for rescue truck. A computer needs a vector algorithm and a sample set of data for calibration of its system. The following example obtains "answers" for one scenario of emergency.

Positions and movements as depicted in texts are kept simple. Motions of real objects occur in four dimensions which are the three dimensions of space and time. The next example takes us a step further; it shows that calculations in three-space are the same as in two-space but with more writing. The task of the example is to locate connection of a wire so dogs can perform.

High Wire Apparatus:   A cubical, steel frame measuring ten meters on each edge is erected in a stadium to support the high wire apparatus for performing dogs. The frame is three dimensional. Perform a calculation to predict a special condition of the apparatus.  

Vectors are powerful, well worth learning. They carry trigonometry, "built-in." What works in two-dimensions works the same way in three-dimensional situations. Also, with time dependence introduced, as say α(t), vector equations differentiate readily to yield relationships of component velocities. The next example, an easy read, treats a three-dimensional situation.  

Pharoh's Engineers  If you use vectors you can watch TV while yop solve this problem about the geometry of the Great Pyramid of Egypt. 

VELOCITY

Consider a body that moves in 0XY-space, for simplicity, in a horizontal plane. Here we introduce a second vector notation: single letters with subscripts. At a specific time, t = t*, the position of the body is P(t*).

The trailing superscript, "0XY" denotes the coordinates of the vector and the subscript associates it with the body. A short time later (at t = t* + Δt) the position of the body is P(t* + Δt). The "difference" of these positions, the second vector minus the first, is expressed implicitly (below left). This difference is a first idea inherent to Newton's Calculus. An increment of time is written: Δt . When the "difference" is divided by its Δt the mathematical symbology is called a "difference quotient" (below right).

Difference                        Difference Quotient:
        

Often we will use a vertical bar as a suffix. The subscript of that bar will identify the system to which the property applies. - the body. The superscript, "0XY" designates the pre-defined reference space of the vector properties of the body (system). The difference quotient is implicit (it does not state something - it implies something. How the quotient changes "in the limiting process" that is, as Δ t vanishes (Δt → 0) can be expressed only implicitly. Abbreviated versions of the quotient that vanishes, are that it equals dP(t*)/dt (the time derivative of position) which also is written as velocity, V(t*). All three notations mean mean velocity.

Position and velocity of the body are consequences of motion, not causes. This is apparent in that they are matter independent. Some argue that momentum is a system property and that position and velocity are characteristics, not properties. Mathematical distinctions are subtle but important. The above discussion used the distinct time, t*, because derivatives are defined at an "instance in time" not in general for all times. That is part of what a derivative is. But once results are in hand, since the chosen t* was arbitrary, any other time might have been chosen. Thus we extend the results, after the fact, to applied to any and all times, t. Each of the three entities (repeated below) is the same thing; each is the velocity of a body in 0XY space.

The above expressions and the ideas behind them were invented by Newton. The center term is called "the derivative of position with respect to tome." It is a vector quantity.

Our next topic is uniform motion. Most readers have "had" uniform motion in high school or university physics. Such coverages tend to avoid vectors, to make things simple with the algebraic formula: S = Vt. Below we revisit uniform motion and in so doing, build upon our calculus. The scenario described below might be a high school physics question. After reading its introduction, can you give it a try on your own. Can you solve Valentino's Wake? Use intuition if you choose.

Valentino's Wake:  From 1920 through 1934, Rudolph Valentino, starred as a dashing hero in over a hundred silent screen movies. But in 1936, at the peak of his popularity, he died. At his wake his body was displayed in an open casket. One newspaper asserted:

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

Is that possible? Could 9,000 people have walked past his casket in an hour? Make some assumptions and prove it is possible or it is impossible?

It is fanciful to ignore math in explaining simple situations. However, simple situations are rare. To address all situations with equal competence requires consistent mathematics which includes vectors. Simple or not, a necessary first step in the solution of uniform motion situations is to select an origin. With Valentino's Wake, Valentino, (in his best clothes in the box) was a natural reference to describe position and motion. In the next case, two boys walk beside a train. Knowing what they know, can you calculate the length of the train?

Trains are constrained to move on tracks and the boys chose to walk parallel to the tracks. Machine mechanisms are constrained by the geometries of the points of connection of their parts. Given a position as with its velocity as input, one can follow that position (and motion) through the mechanism constraints to tell what the output position and velocity will be. These calculations are not fun; some persons get paid well to put up with that. Below is a question about design of a scissor jack. There is a scissor jack in the trunk of your car; we hope it stays there. 

Scissor Jack:   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) 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. Suppose, (in the position shown - dimensions given below), the driver rotates member BC at = 200 revolutions per minute. Calculate the upward velocity of the top of the jack, point D.

MOMENTUM

Pen in hand, pages before statement of his laws, Newton defined "quantity of matter" (the physical property of a system; its mass). Next he took mass, multiplied it by its velocity and called the vector result, a "quantity of motion." Thus married Newton, the physicality of a body (mass) to its time-wise existence in space - velocity. We call that time dependent vector product, momentum. Newton applied calculus to momentum in the same fashion as previously was done to position. Position being spatial, its derivative was velocity, also spatial. Momentum is material (space and matter). Newton decided the derivative of momentum must equal something new. It would equal whatever might change momentum - Newton named those effects - forces.

We will call this equation: the momentum equation for a body. Left of equality represents the potential time rate of change of momentum of the body in its space, 0XZ. Right of equality are the "change agents," the causes of whatever change of momentum might occur. These effects of the surroundings are called forces. Were momentum to change, force is the reason.

In mathematics the above equation is classed as a first order vector differential equation. The equation is implicit; it applies to all cases of motion of a body. Answers to real physical problems are explicit, not implicit. The above equation is applied to a real situation by sppecification of the initial state of the body and the manner of action of forces over the event of the body. Given that information, the implicit momentum equation is integrated to yield an explicit answer. This will be demonstrated repeatedly in what follows.

A final comment fits in well with what we will do next. We prefer the notation ΣF to the simpler F. Usually more than one force participates in momentum change of a body. Students taught to use F often use one force, the first force they find. Σ is a math symbol meaning "the sum of all" Fs.

It would be nice now to solve some easy and interesting-to-learn examples by use of the momentum-equation. Easy problems are not easy to find.

Practicing the Basketball Chest Pass:   One manner of passing a basketball is called the "chest pass." Players practice this by making crisp passes of the ball into a spot on a wall of the gym. The ball experiences change of momentum. Here we calculate the change of momentum one ball experiences. Also, a step further, we use the event to provide an idea of pressure.

P-51 MUSTANG:    As a second problem related to momentum we consider the actual "loss of speed in straffing" that WWII P-51 Mustang pilots experienced. To fly low to the ground then be slowed as a consequence of the streams of lead of their machine-guns could get a pilot shot down or worse. Straffing took courage. The courage you need next is to set up initial conditions of momentum for a system with two masses, then solve a first order differential equation. Some approximation will be needed, as usual.