CHEMICAL BONDING _see Chapter 9 for more detail_ by dfhercbml


(with a small amount of additional material)

In this chapter, we will discuss the basic laws and
concepts of “classical physics”: motion; vectors; force;
Newton’s laws of motion; Newton’s law of universal
gravitation; and momentum and collisions. In a more
traditional physics course for science majors, a great
deal of mathematics (including calculus) would be
required and utilized. In this course, while a little math
will be employed, it will be kept to a minimum.

Not all of the material in this chapter will be covered.
Rather, the focus will be on that which is necessary to
address your questions.

An Historical Note
The first significant attempt to develop a physical theory
of motion was due to Aristotle. Though his view of
motion was seriously flawed, it was widely accepted for
nearly 2000 years.

Basically, it suggested that forces on Earth were either
“natural” (e.g. “gravity”, which explains why an object
falls) or “unnatural or violent” (e.g. wind, a human

Among the flaws in Aristotle’s view:
 1. As well as a natural force of “gravity”, he proposed
    a natural force of “levity” (which could cause smoke
    to rise into the air). We know now that this is really
    a result of gravity.

  2. Aristotle’s forces were all localized to Earth, and
     could not explain “universal” motion, like that
     observed in the heavens.

It was not until the 1500s and 1600s AD that significantly
different theories of motion were proposed, thanks to the
work of Galileo, Newton and others, with Newton’s work
being elegant, groundbreaking and still widely useful to
this day.

In last century, we have come to understand that
Newton’s theory, while broadly valid in explaining most
motions observed here on Earth, is itself flawed and
limited in its scope. Specifically, while Newton’s laws
continue today to provide accurate predictions of many
“everyday” motions, other motions involving extremely
high speeds or extremely large masses are currently best
explained using Einstein’s relativity theory, while still
other phenomena best fit Quantum physics.

As we have pointed out before, a “theory” is just that. It
can never be proved right, and may someday be proved
wrong, or at least (as is the case with Newton’s laws) be
shown to have only limited application.

Motion is obviously detected as some sort of spatial
change over time. That is, without the ability to measure
and record length (or distance) and time, we would be
unaware of motion, or at least unable to describe it in
any concrete, quantitative way. So, as we will see, all of
the quantities used to describe motion involve space
and/or time.

Fundamental Concepts of Motion
Here are the basic concepts used in detecting, observing
and describing motion.

  1. Position (or location)
     Since motion must involve a change in position, this
     is the most basic concept of all. In order to describe
     the location or position of an object, we need some
     sort of coordinate system (e.g. Cartesian coordinates
     in 2- or 3-dimensions, a common street or road map,
     latitude/longitude on the Earth’s surface, etc.). For
     the purpose of this discussion, we can use a letter
     (e.g. x, y, z, r) to represent a body’s (or a particle’s)
     position. Position is obviously measured in units of
     length (m, cm, km, etc.)

2. Displacement (or change in position)
   Displacement is just the difference between a body’s
   (or particle’s) position at an initial time (or at an
   earlier time) and its position now (or at a later time).
   In mathematics, we often use the “delta” symbol  to
   depict “change”, so for example, we could write
                    x  x f  x i
  where xf and xi stand for the final (later) and initial
  (earlier) position coordinates. Like position,
  displacement is also measured in units of length.

  For example, suppose we use positive and negative
  numbers to represent positions east and west of
  North Bay, respectively. Then Mattawa and
  Sudbury might be located at (say) x = +55 km and x
  = –125 km. If you drove from Sudbury to Mattawa,
  your displacement would be
       x  55   125   180 km (i.e. 180 km east),
  while driving from Mattawa to Sudbury would
  produce a displacement of
      x  125   55   180 km (or 180 km west).
  Note that the formula automatically distinguishes
  east and west by assigning the appropriate  sign!
  In other words, displacement has an associated
  direction attached to it.

3. Distance
  Distance is the length of any path traversed as an
  object moves. We often use the letter d for distance,
  as in the earlier formula d = ct used to describe the
  distance traveled by light (or radio waves) through
  space over time. Again, distance is measured in
  units of length, but, unlike displacement, distance
  has no associated direction.

  It might be tempting to think, otherwise, that
  distance and displacement are the same. In fact,
  they are if the motion takes place in a straight line
  and in one direction, but otherwise they are

  For example, consider a body that starts out at point
  P and, after traveling for some time, eventually
  returns to that same location.


  In this case, the distance d would be the total length
  of the the “squiggly curve”, but the displacement
  x = 0! (In fact, any “closed path” that ends where
  it begins would have nonzero distance but zero

  displacement.) Thus, while “distance” and
  “displacement” are sometimes used interchangeably
  in everyday conversation, they have different
  meanings in physics!

4. (Average or Instantaneous) Speed
   Speed is “how fast” a body travels, or the rate at
   which distance is covered. If we denote speed in
   general by v (we used c earlier when it referred
   specifically to the speed of light), then we have
   already seen that distance d, elapsed time t and
   speed v are related by
                 d (in m/s, km/h, etc.).
  Alternatively, of course, this equation can be
  rearranged into the equivalent forms
                  d  vt, t 
  If the speed is calculated over a “reasonable” length
  of time (e.g. several seconds or longer) we think of it
  as an average speed over that elapsed time. On the
  other hand if you imagine a time of a mere fraction
  of a second, then the speed will essentially become
  instantaneous. (The speedometer in a car indicates
  the speed at any moment in time, and is thus
  showing instantaneous speed.)

5. (Average or Instantaneous) Velocity
   In the same way that displacement is somewhat like
   a distance, velocity is somewhat like speed. In
   particular, velocity has an associated direction. The
   easiest way to incorporate this into the definition is
   to define velocity in terms of displacement (which
   has a direction), rather than in terms of speed. So,
   we can define velocity (also denoted by v) by velocity
   = displacement/time interval, or
              x x f  xi
          v                  (in m/s, km/h, etc.)
              t    t f  ti
  Returning to our Mattawa-Sudbury example, if you
  left Mattawa at 1:00 PM and arrived in Sudbury at
  3:00 PM, your average velocity would be
        x f  xi        125  55    180
   v                                    90 km/h (west).
        t f  ti            2           2
  As with speed, if the time interval t was imagined
  to be extremely small, i.e. was approaching zero,
  then the average velocity would become
  instantaneous velocity.

  Since velocity always has an associated direction
  while speed does not, we see that, in physics, “speed”
  and “velocity” are different, even though the two
  words are often used to mean the same thing in
  ordinary conversation.

  In particular, as we are about to see, an object can
  have a changing velocity even though its speed is

6. (Average or Instantaneus) Acceleration
   Even though we often associate acceleration solely
   with a change of speed, this is not really the case.
   Formally, acceleration is defined as the rate of
   change of velocity, i.e.
         v v f  vi
      a             (e.g. in m/s per s, or m/s2).
         t t f  t i
  It is especially important to note that, since velocity
  always has an associated direction, then velocity can
  also change when the direction of motion changes
  (even if the speed stays the same)!

  Think of a car which is turning a corner (or
  negotiating a curve) at constant speed. As it travels
  around a curved path, its velocity changes direction.
  Therefore, it experiences a velocity change v, and
  so it accelerates.

  In exactly the same way, a body traveling in a
  (nearly perfect) circle at (nearly) constant speed,
  such as the Moon as it orbits the Earth, is actually
  accelerating. This may take some getting used to, if
  you are accustomed to associating acceleration with
  a change of speed rather than a change of velocity!

Just as for speed and velocity, an acceleration will be
an average acceleration if the change in velocity
occurs over a significant time interval, but becomes
instantaneous acceleration if the time interval
shrinks to zero.

As one example, suppose you start from rest, and
begin moving east until you reach a final velocity of
10 m/s (east) five seconds later. Then your average
acceleration is
                 v f  vi        10  0
          a                             2 m/s2 (east).
                 t f  ti          5
For another example, imagine a car whose brakes
are applied, slowing it from 24 m/s (east) to 6 m/s
(east) during a 6-second interval. Then, its average
acceleration is
      v f  vi   6   24    18
 a                            3 m/s2 (west).
      t f  ti       6           6
(Note, in this case, since the car is slowing down, that
the acceleration is actually directed opposite to the
car’s motion!)

Shortly, we will see that the acceleration associated
with an object traveling around a circle at constant
speed is actually directed toward the centre of the
circular path.

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