1
Massimiliano Galeazzi: Motion in 1 dimension – Any Reproduction or distribution without the author’s consent is forbidden
.
Motion in 1 Dimension
By Prof. Massimiliano Galeazzi, University of Miami
When you throw a pebble straight up, how high does it go? How fast is it when it gets back?
If you are in your car at a red light and the light gets green, how long does it take you to reach the speed limit? How
far have you traveled?
These are just some of the questions that we will be able to answer in this chapter. The focus of the chapter is
motion in 1-dimension, that is, we will only consider objects moving along a straight line, which could be horizontal, like
in the example of the car, vertical, like in the example of the pebble, or have any other orientation.
In this chapter we will also neglect the physical shape of the object, for example the fact that the pebble could also
rotate, or that the car has many parts moving in different ways. In general, we will refer to all objects a
point particles
, or
simply
particles
, meaning that their physical geometry is negligible compared to the motion we are describing.
---------------------------------------------------------------------------------------------------------------------------------------------------
MATH INSERT
The description of this chapter is completely based on calculus and the use of derivatives. Since some of the
students reading this chapter are also taking a parallel calculus course, we will start with an overview of the calculus
principles that we will need. This is by no means intended as an exhaustive treatment of functions and derivatives, but just
a functional description in support of this chapter, mostly for the benefit of students that have not seen it yet, but will
cover it, in a much more formal way, in the next few weeks.
2.1 Functions and variables
In high school we have learned that
ݕൌ݂ሺݔሻ
or
ݕሺݔሻ
is used to indicate that the variable
ݕ
is a function of the
variable
ݔ
(i.e., the values of
y
depend on the corresponding value of
x
). The function may have many forms and describes
how the variable
y
changes as a function of the variable
x
.
Before proceedings it is important to remember that
y
and
x
are just two of the many symbols that can be used in
functions and their preferred choice in many math problems is simply due to the general convention of using
x
and
y
as
canonical variables for the Cartesian plane, but we can as well write
ܽൌ݂ሺܾሻ
to describe the fact that the variable
a
is a
function of the variable
b
. In particular, in physics we tend to assign different variables to different quantities to avoid
confusion between them, as there are many different quantities that will be introduced in this book alone. So many, in
fact, that we will make use of the full Latin and Greek alphabets, both lower and upper case, and we still won’t have
enough symbols to describe all the physical quantities that we will introduce. From the mathematical point of view, this
means that you should get used to see function such as
ߙൌ3ܿ
ଶ
, which simply means that the quantity
ߙ
(the Greek letter
alpha) depends on the quantity
c