# CHEMICAL BONDING _see Chapter 9 for more detail_ by dfhercbml

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```									CHAPTER 5: MOTION AND GRAVITY
(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

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
“push”).
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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.
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DESCRIBING MOTION
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.)
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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,
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.
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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
different!

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

P

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
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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.).
v
t
Alternatively, of course, this equation can be
rearranged into the equivalent forms
d
d  vt, t 
v
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.)
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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.
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In particular, as we are about to see, an object can
have a changing velocity even though its speed is
constant!

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!
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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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