Chapter 2 - Motion in One-Direction
Definitions of words you'll hear a lot…
 Frame of reference - before solving any physical problems, you
need to consider the frame of reference. The frame of
reference is a choice of coordinate axes that defines the
starting point for measuring necessary quantities. I'll often say
"pick where your origin will be" and "choose a positive direction";
each of these things make up your frame of reference.
 Displacement - the displacement of an object is defined as its
change in position. We will use x to represent the displacement
of an object. This is not the same thing as the distance traveled.
displacement = the change in position
x  x f  x i
Displacement is measured in meters (m).
 Vector - a vector is a quantity that is characterized by both a
magnitude and a direction. Vectors are denoted in bold font or as


letters with an arrow over the letter. (Example: v or a )
 Scalar - a scalar is a quantity that is characterized by only
magnitude, not direction. (Example: time or speed)
 Average Speed - the average speed of an object is defined as
the total distance traveled divided by the total time elapsed.
average speed = total distance/total time
Average speed is a scalar quantity and it is measured in meters
per second (m/s).
 Average Velocity - the average velocity of an object is defined
as the displacement divided by the total time elapsed. We will
use v to represent average velocity.
average velocity = displacement/total time
v
x x f  xi

t
t f  ti
Average velocity is a vector quantity and it is measured in meters
per second (m/s).
 Instantaneous Velocity - the instantaneous velocity of an object
is defined as the limit of the average velocity as the time interval
t goes to zero. We will use v to represent instantaneous
velocity. We will usually be lazy and refer to this as "the
velocity".
x
t  0  t
v  lim
Instantaneous velocity is a vector quantity and it is measured in
meters per second (m/s).
You drive from here to Atlantic City. The total distance traveled is 70
miles. You drive the first 20 miles at 55 mph and the next 45 miles at
65mph and the last 5 miles at 45 mph. The total trip takes you 1 hour
and 30 minutes.
What is your displacement?
What is your average speed?
What is your average velocity?
You're done in Atlantic City and you drive back home. You travel
another 70 miles but this time you drive the first 10 miles at 40 mph,
the next 50 miles at 70 mph, and the last 10 miles at 35 mph. This
time, the trip takes you 1 hour and 15 minutes.
What is your total displacement for your entire trip?
What is your average speed?
What is your average velocity?
 Average Acceleration - the average acceleration of an object is
defined as the change in velocity divided by the total time
elapsed (the change in time). We will use a to represent average
acceleration.
average acceleration = change in velocity/change in time
a
v v f  vi

t t f  ti
Average acceleration is a vector quantity and it is measured in
meters per second per second (m/s2).
 Instantaneous Acceleration - the instantaneous acceleration of
an object is defined as the limit of the average acceleration as
the time interval t goes to zero. We will use a to represent
instantaneous velocity.
v
t  0 t
a  lim
Instantaneous acceleration is a vector quantity and it is measured
in meters per second per second (m/s2).
Graphical Representations of Position, Velocity, and Acceleration
This is called a Position vs. Time Graph
The average velocity of an object during the time interval t is equal
to the slope of the straight line joining the initial and final points on a
graph of the object's position vs. time.
Slope = change in vertical axis/change in horizontal axis
Slope = rise/run
Slope =
x
v
t
The slope of the line tangent to the position vs. time curve at "a given
time" is defined to be the instantaneous velocity at that time.
Similarly, the slope of a Velocity vs. Time Graph is defined as the
average acceleration. And the slope of the line tangent to the velocity
vs. time plot at a given time is defined as the instantaneous
acceleration.
You can plot Velocity vs. Time and Acceleration vs. Time Graphs too.
Match the velocity vs. time graphs to the appropriate acceleration vs.
time graphs.
Graph (a) has a constant slope, indicating a constant acceleration,
which is represented by (c).
Graph (b) represents an object with increasing speed, but as time
progresses, the lines drawn tangent to the curve have increasing
slopes. Since the acceleration is equal to the slope of the tangent line,
the acceleration must be increasing and the graph that best indicates
this behavior is graph (d).
Graph (c) depicts an abject that first has a velocity that increases at a
constant rate, which means the acceleration is constant. The velocity
then stops changing, which means the acceleration of the object is
zero. This behavior is best matched by graph (f).
NOTE:
When there is no change in the velocity of an object, the acceleration
of the object is zero.
Is there such a thing as a negative acceleration? Yes. If the object is
slowing down, then the object is said to have a negative acceleration
(deceleration is not a word we use in physics).
Motion Diagrams - A motion diagram is a representation of a moving
object at successive time periods, with velocity and acceleration
vectors sketched at each position. A motion diagrams looks like a
series of photographs of the objects. The time intervals between
"photographs" in the motion diagram are assumed to be equal.
One-Dimensional Motion with Constant Acceleration
When an object moves with constant
acceleration, the instantaneous
acceleration at any point in a time
interval is equal to the value of the
average acceleration over the entire
time interval.
So, the average acceleration equals the
instantaneous acceleration when a is
constant.
a
v f  vi
t f  ti
We can, for convenience, decide that
t i  0 . We can also let vi  vo (the initial
velocity at t = 0) and let v f  v (the
velocity at any time t)
a
v  vo
t
or
v  vo  at
The graphical representations of motion with constant acceleration
show that the velocity varies linearly with time. Because this change is
linear (or uniform, using physics terms) over time, we can express the
average velocity in any time interval as the average of the initial and
final velocity:
v
Since
vo  v
2
x x

t
t
v v
x  v t   o
t
 2 
v
or
x  12 vo  v t
We can substitute v  vo  at in the above equation to get a different
form
x 
1
2
vo  vo  at t  12 2vo  at t  12 2vo t  at 2 
x  v o t  at
1
2
2
We can solve for one more very useful equation by solving v  vo  at for
t and plugging the result into x  12 vo  v t
2
 v  vo  v  v o
x  vo  v 

2a
 a 
2
1
2
or
v  vo  2ax
2
2
Example: A car traveling at a constant speed of 24 m/s (not 45 m/s as
indicated in the picture) passes a hidden state trooper on the side of
the road. One second after the speeding car passes the trooper, he
sets off in chase with a constant acceleration of 3 m/s2. (a) What is
the acceleration of the speeding car? (b) How long does it take the
trooper to overtake the speeding car? (c) How fast is the trooper
going at that time?
Freely Falling Objects
A freely falling object is any object moving freely under the influence
of gravity alone, regardless of its initial motion.
Objects thrown upward or dropped (or thrown) downward are all
considered freely falling.
We will denote the magnitude of the free-fall acceleration (gravity) by
the symbol g . The value of g varies with altitude and it varies
slightly with latitude; however for our purposes, we will consider the
value of g as a constant.
g  9.80 m/s2
Since g is a constant acceleration, we can use all of the equations we
derived for motion in one direction with constant acceleration. It is
conventional to define "up" as the +y-direction and to use y as the
position variable. In that case, the acceleration is
a   g  9.80 m/s2