Name__________________________Period____
Uniform Motion - Reading 1
In physics we attempt to describe the natural world using mathematical models. We will begin
our physical investigations with a study of motion. The most basic type of motion to model is uniform
motion, that is, motion that occurs at a constant speed or velocity. Before we delve into our investigations,
we must make our mathematical knowledge slightly more robust. To do so, we will introduce the concept
of a vector and see its application in physics. You should keep this reading as a guide for the remainder of
the unit.
Vectors and Scalars:
Vectors are among the most powerful and important tools we have in physics to analyze many
different natural phenomena. So what are they? A vector is much like the mathematical equivalent of a
unit. Much like the unit of the meter defines a certain length in physical space, a vector defines a "length"
in a mathematical space. For this reason, vectors are often used to model physical quantities such as
lengths, velocities, accelerations and many other things. Before we see how, we must get familiar with
what a vector is made up of.
A vector has two pieces: a magnitude and a direction. The magnitude describes the "length" of
the vector (or more generally, the numerical amount the vector symbolizes). The direction describes
which way the magnitude points in. This raises the question, how can a magnitude point in a direction?
Well, in math class vectors exist in the fictional "Cartesian space" and can point in the 𝑥̂ direction or the
𝑦̂ direction. Note the use of vector hats on the x and y to symbolize that I am referring to the x and y
directions, not the variables. Some examples are shown below
y
3𝑦̂
1.5𝑦̂
6𝑥̂
-6𝑥̂
x
Notice that each vector in the Cartesian plane (x-y axes) above is labeled with two pieces of
information: a magnitude (the number representing the length of the vector) and a direction (an 𝑥̂ or 𝑦̂ to
indicate if the vector lies parallel to the x or y axis). Also notice that the magnitude is not necessarily
related to the vector's position on the Cartesian plane. Each vector can be slid around on the plane because
the vector is defined by its magnitude and direction, not its position on the plane. Another convention
worth noticing is that vectors with a positive magnitude point in the same direction as the given axes,
whereas vectors with a negative magnitude point in the direction opposite to the given axes.
Now that we know something about vectors, we can examine which physical quantities can be
modeled using vectors and which cannot. In general, any physical quantity that has a magnitude and a
direction can be modeled with a vector. Some examples include:
- Velocity: you can drive at 25mph South or 25 mph North
- Force: you can push a box with 10 pounds of force to the left or to the right
Any physical quantity that does not have a direction can only be modeled as a scalar. A scalar
has a magnitude only. Examples of scalars include:
- How much money you have: you cannot have $24 up or $24 down
- Mass: An object might weigh to be 20kg. It cannot be 20kg left or 20kg right
Definitions for Uniform Motion:
Finally! we know enough to make some careful definitions of everyday terms that we might already be
misusing. What do I mean? Well, we all use words to describe motion in everyday life; words like
position, location, time, speed, velocity. But in physics class, we need to define what these words mean
very narrowly if we are to avoid confusion, so let's get to it!
Term
Symbol(s)
Vector or
Definition
Scalar?
Position
x
Scalar
The physical location of an object as measured from some
y
(typically)
zero point (such as Mr. J is 2 meters away from the door)
Displacement
Δx
Vector
The difference between initial position and final position
Δy
Δx = xfinal - xinitial
Clock reading
t
Scalar
The clock reading at which an event occured
Time interval
Δt
Scalar
The amount of time that passes between two events
Δt = tsecond event - tfirst event
Speed
speed
scalar
Either: 1) the magnitude of velocity
Or
2) the amount of distance traveled divided by the
time period
Velocity
V
Vector
The speed an object travels at and the direction it is
traveling in:
v = Δx/ Δt
Here I will also introduce the three formulae that will be provided on the unit test. We will dig into where
they come from later, but for right now read them over and see if you understand their translations:
xf = xi + 𝑣̅ Δt
The final position of an object equals the initial position plus the average velocity at which the object was
traveling times the amount of time it was traveling for.
∆𝑥
𝑣̅ = ∆𝑡
The average velocity of an object is equal to the displacement the object covered in a time interval
divided by the time interval
𝑃𝑎𝑡ℎ 𝐿𝑒𝑛𝑔𝑡ℎ
speedaverage =
∆𝑡
The average speed of an object equals the path length of the object divided by time interval it took to
traverse that path length.
The difference between speed and velocity seems tiny, but the difference shall become important in time,
for right now, take a look at the following example that illustrates a time when two runners can have the
same velocity but different speeds. Suppose "runner A" and "runner B" run the two paths shown below.
Δxrunner A
Each runner completes their run in 3 seconds
Start
A
A
B
B
Finish
Δxrunner B
Notice how since the displacement Δx is the same for runner A and runner B, they will have the
same velocity since both trips took 3 seconds. However, runner B had a greater path length, so they
would have a greater speed.