1-Dimensional Kinematics
I. 
Kinematics
• 
• 
The word kinematics is used to identify the science of the study of motion.
Kinematics falls in under the field of Mechanics. Mechanics covers a
broad range of topics that cover the study of, and modeling, of the motion of
objects.
Kinematics uses models, graphs, equations, numbers, to help describe the motion
of objects.
• 
II.  Scalars
• 
• 
• 
• 
• 
A scalar is a quantity that is described only by its magnitude (size), using a number
The length of this room is a scalar quantity.
The product of the length and width of this room (area) is a scalar quantity.
The speed at which you ran to get to my class is a scalar quantity.
A scalar quantity will not indicate a direction. For example, you may push on a box with
50 N of force, but this number and unit do not reveal in which direction you pushed.
III. Vectors
• 
• 
• 
A vector is a quantity that is described by both magnitude and direction.
Direction includes a description of which way the magnitude of the vector
When you push a box with 50 N of force 40o N of E, you’ve described a vector quantity
1
Below is a partial table of quantities defined by physicists as either scalar or vector. The
use of the proper word is important. For example, speed and velocity are not the same
thing. Make sure you learn this table and take care in how you use terminology.
2
IV. Displacement and Distance
•  Displacement is a VECTOR. It describes how far and in what direction an object has
moved from its starting point.
•  Distance is a SCALAR. It describes the length an object has moved. The SI units are
meters, m
•  Over very SHORT distances, the displacement scalar and distance are the same.
•  Look at the diagram below. An ant has crawled quite a long distance yet was not very
far displaced from its starting point.
stop
distance
displacement
starting point
•  Displacement can be thought of as a change in position whereas distance is the
amount of ground covered
•  Displacement is the shortest path between two points so displacement will never
3
be larger than distance
V. Speed and Velocity
A.  Average Speed and Instantaneous Speed
•  Speed is a scalar. It tells you the rate at which you cover a certain distance.
•  Speed is the rate at which an object covers a given distance.
•  If you live 15 miles from school, and it takes you 30 minutes to get there, your average
speed is the ratio of the total distance (15 miles) and the total time (0.5 h). Your
speed was 30 miles/h
Δs
v=
Δt
Speed is directly proportional to
distance and inversely proportional to time
v is speed
Δs is change in distance typically measured in meters
Δt is change in time typically measured in seconds
• 
Instantaneous speed is the speed at a given instant of time. It can be approximated
by dividing distance traveled over a very short time interval. In other words, Δt should
be vanishingly small to get a good value for instantaneous speed.
• 
The average speed is the average of all instantaneous speeds over a given time interval.
4
The table below summarizes how average speed and instantaneous speed are related
distance (mi)
0
0.1
1
5
7
8
9
11
14
15
time (h)
0.05
0.10
0.15
0.20
0.25
0,30
0.35
0.40
0.45
0.50
speed (mi/h)
0
2
18
80
40
20
20
40
60
20
Δs
(1 − 0.1)mi
v=
Δt
=
(0.1 − 0.05)h
=
0.9mi
mi
= 18
0.05h
h
•  From the table above, average speed is 15 mi/0.5 h = 30 mi/h
•  Averaging the approximate instantaneous speeds in the third column, the average speed
is 30 mi/h.
5
B. Velocity
•  Velocity is a vector.
•  The SI units for velocity are m/s
•  Velocity is the ratio of the scalar of the displacement vector and time.
•  The scalar of velocity will never be larger than the value for speed because the displacement
scalar is never larger than distance
•  On very SHORT distance intervals, displacement and distance are essentially the same.
•  On very SHORT time intervals, speed and the scalar of velocity are essentially the same.
•  The instantaneous speed of an object is equal to the scalar of its velocity.
 Δr
v=
Δt
v is speed
Δr is the magnitude of the displacement vector typically measured in meters
Δt is change in time typically measured in seconds
•  Velocity is reported with a direction
•  At constant speed, velocity is constant UNLESS the object changes direction
•  A car going around a curve at 5 m/s has a constant speed but a changing velocity
6
C. SUMMARY of SPEED and VELOCITY
• Speed is the rate of change of distance with time.
• As a scalar it has magnitude only.
• Average speed is speed measured over a non-zero time interval.
• Instantaneous speed is estimated over very small time intervals.
• The symbol for speed is v (italic).
• Velocity is the rate of change of displacement with time.
• As a vector it must be stated with both magnitude and direction.
• Average velocity is velocity measured over a non-zero time interval.
• Instantaneous velocity is estimated over very small time intervals.
• The symbol for velocity is v (boldface) or v with an arrow over it.
7
VI. ACCELERATION
•  Acceleration is a vector. It is described with both a magnitude and a direction.
•  Acceleration is the change in velocity with respect to time.
•  The SI units for acceleration are m/s2
•  Look at the units for acceleration. Can you see the units for velocity “hidden” there?
•  The units are telling you that acceleration is the rate of change of velocity
•  A change in the object’s direction is a change in acceleration (even if speed is constant)
•  A change in velocity by speeding up or slowing down is a change in acceleration
 Δv
a=
Δt
Note that acceleration is
directly proportional to velocity
and inversely proportional to time
a is acceleration
Δv is the magnitude of change in the velocity vector
Δt is change in time typically measured in seconds
8
VII. Basic Application of Vectors
DIRECTION: + direction means to the right or up
- direction means to the left or down
A vector with a magnitude of -20 units has the same magnitude as a vector with
+20 units. The signs just tell you the vectors point in opposing directions
•  Vectors are represented in diagrams with arrows. The length of the arrow tells you
the magnitude of the scalar value and the arrow points in the direction the vector is acting
the arrow is a velocity vector that indicates the direction and speed of
the ball. The shorter arrow in the next diagram indicates that the speed
of the ball is decreasing. The ball IS accelerating but in a - direction
The white arrow pointing to the left is the acceleration vector.
Generally, when an object slows down, the acceleration vector points
in a direction OPPOSITE of the object’s motion.
9
Let’s drop an object from an airplane…
among other things (to be discussed later), gravity acts on the object and it
accelerates it as it falls
The diagram shows how we might represent this acceleration due to gravity. The velocity
gets larger, thus the object accelerates.

a

v
The arrows point downward to
indicate the direction of velocity
and they get longer to indicate that
the object is increasing in speed.
Acceleration and velocity point in the - direction
acceleration is constant
(vector arrow has one length)
velocity changes
(vector arrows change length)
10
A ball tossed up slows down…
…but its velocity increases on the trip down

a

a

v

v
acceleration is constant
(vector arrow has one length)
velocity changes
(vector arrows change length)
The acceleration vectors point down (- direction) on the trip up and the trip down
(decelerates going up, accelerates coming down)
11
VIII. Graphical representation of Velocity and Acceleration
A. Position vs. Time Plots
If you made a plot of position vs elapsed time for a moving object, what information
would you get for your trouble?
•  Visualize velocity
•  The shape of the plot tells you if velocity is constant, changing or if the object is moving at all
•  The slope of the line at a given point gives instantaneous velocity
 Δr
v=
Δt

Δr = vΔt
rearrange…
b=0
y = mx + b
can you see from the equation
above that a plot of position vs. time
would give velocity as the slope
12
A plot that is linear, with a positive slope,
indicates that the object is moving
with a constant velocity in the + direction
The steeper (larger) the slope, the higher
the velocity
A plot that is linear, with a negative slope,
indicates that the object is moving
with a constant velocity in the - direction
The steeper (larger) the slope, the higher
the velocity
The negative slope does NOT mean the object is slowing down…it means the
displacement of the object from its origin is getting smaller (sort of like it is going
to its home position)
13
Position vs. time plots with curvature
Curvature indicates a CHANGE in velocity (the slope of the line is changing!) and
therefore an accelerating object
+ direction velocity
starts slow, speeds up
if you want to know if the object
is speeding up or slowing down,
look at how the slope changes
over time…the steeper it is, the
higher the velocity
-  direction velocity
starts fast, slows down
-  direction velocity
starts slow, gets faster
14
B. Velocity vs. Time plots
If you made a plot of velocity vs elapsed time for a moving object, what information
would you get for your trouble?
•  Visualize acceleration
•  The shape of the plot tells you if the acceleration of the object is constant or changing
•  The slope of the line at a given point gives acceleration
higher v
+ direction
- direction
the – sign does
NOT mean slowing
down…it just indicates
direction
line means v = 0
 Δv
a=
Δt

Δv = aΔt
rearrange…
b=0
y = mx + b
can you see from the equation
above that a plot of velocity vs. time
would give acceleration as the slope
15
C. Acceleration Graphs
a plot of velocity vs. time gives a profile of the acceleration of an object.
+ acceleration
when I’m speeding
up I’m moving
away from
v = 0 with time
- acceleration
+ acceleration
- acceleration
+ acceleration
- acceleration
- acceleration
when I’m slowing
down I’m moving
toward v = 0 with time
+ acceleration
16
no acceleration, slope = 0
velocity + direction
•  IF THE PLOT IS MOVING TOWARD V =0
WITH TIME, THE OBJECT IS SLOWING
DOWN.
•  IF THE PLOT HAS NO CURVATURE OR
CHANGE IN SLOPE, ACCELERATION IS
CONSTANT
If the plot crosses v =0,
then the object has
CHANGED direction
The AREA under a v vs. t graph
represents the displacement
of the moving object
Graphs copied from The Physics Classroom on-line
17
IX. Gravity and Free Fall
A.  GRAVITY
•  Gravity is classically considered a force of attraction between two or more objects.
The more massive the object, the more gravitational force it has.
•  gravity causes an object to accelerate downward. The value of this acceleration is
constant near the surface of the Earth
acceleration due to gravity near the Earth’s surface:
m
g = −9.8 2
s
the negative sign just indicates the DIRECTION OF THE
acceleration vector…DOWN
18
object dropped
B. FREE FALL
t is the time in free fall
y is the distance of fall
v is the velocity
look how the velocity increases
with time. Look at how the
length of the velocity vector changes
The object accelerates by nearly
10 m/s every second
19
B. FREE FALL
•  an object in FREE FALL is, by definition, moving only under the influence of gravity
•  Objects in free fall accelerate at the same rate, regardless of mass
•  an object in free fall speeds up as it falls. Its velocity changes by 9.8 m/s every second
until terminal velocity is reached
Graphs of free fall
curvature indicates that the
velocity is NOT constant (slope
is changing with time)
notice that the object has constant
acceleration in the negative direction…
it speeds up as it falls
20
X. The 1-D Kinematic equations
The following equations relate all the stuff we’ve been talking about in these notes
See why math is so wonderful? The whole unit is contained in the 4 equations below!
THE KINEMATIC EQUATIONS
1 2
s = vi t + at
2
2
2
v f = vi + 2as
v f = vi + at
s=(
vi + v f
2
)t
The i and f subscripts mean initial and final values
ASSIGNMENT: Convince yourself that these
equations make sense by plugging the appropriate
units into the right hand side of the equation.
Make sure the units work out properly to match
the variable on the left hand side of the equation.
21