Unit 2 1-Dimensional Kinematics
• Kinematics --the science of describing the motion of objects
• Methods --using words, diagrams, numbers, graphs, and equations
• Terms -- scalars, vectors, distance, displacement, speed, velocity and
acceleration
I Vector Analysis
Objectives:
• What is vector quantity
• How to identify a physics quantity as scalar or vector type.
• How to graph vector
• How to add or subtract vectors in 1 dimensional space
1 Scalars and Vectors
1. Definition
A scalar quantity is one that can be described by a number with proper
unit (magnitude):
Temperature (such as-5℃)
Speed (such as 5m/s)
Length (height as 10m)
mass ( such as 2 lb, or 2kg)
energy and power ( joule, watts)
density(1kg/m3)
time, time duration,
distance
A vector quantity is a quantity that is fully described by both magnitude
and direction.
velocity, acceleration, force, displacement
Distance moved is the actual length of the path along
which an object travelled. It is a
scalar.
Displacement is the straight line distance from the
beginning to the end of the path
along which an object moved. It is a vector and has
nothing to do with how the object
got there.
2. About Vector:
Vector quantity demands that both a magnitude
and a direction are listed.
Example:
If say, "A bag of gold is outside the classroom. To find it,
displace yourself 20 meters." --not enough information to
find the gold.
If say, "A bag of gold is located outside the classroom.
To find it, displace yourself from the center of the
classroom door 20 meters in the west direction.“ –you
will find it.
Examples of Vector Quantities:
• I travel 30 km in a Northerly direction (magnitude is 30
km, direction is North - this is a displacement vector)
• The train is going 80 km/h towards Sydney (magnitude
is80 km/h, direction is 'towards Sydney' - it is a velocity
vector)
• The force on the bridge is 50 N acting downwards (the
magnitude is 50 Newtons and the direction is down - it
is a force vector)
Each of the examples above involves magnitude and direction
Check vector and scalars:
http://www.physicsclassroom.com/class/1DKin/Lesson-1/Scalars-and-Vectors
3. Vector Notation
• Some book use a bold capital letter to name vectors. (example, a
force vector F).
• Some textbooks write vectors using an arrow above the vector
name, like this: 𝐹
• In Graph:
A vector is drawn using an arrow. The length of the arrow indicates
the magnitude of the vector. The direction of the vector is
represented by (not surprisingly :-) the direction of the arrow.
𝐴
𝐴
Not location
related
Example 1 - Vectors
vector A has direction 'up' and a magnitude of 4 cm.
Vector B has the same direction as A, and has half the magnitude (2 cm).
Vector C has the same magnitude as A (4 units), but it has different direction.
Vector D is equivalent to vector A. It has the same magnitude and the same
direction. It doesn't matter that A is in a different position to D - they are still
considered to be equivalent vectors because they have the same magnitude and
same direction. We can write:
A=D
Note: We cannot write A = C because even though A and C have the same
magnitude (4 cm), they have different direction. They are not equivalent.
This is called scaled vector
diagram.
Scale: 1cm=1km
4
A zero vector has magnitude of 0. It can have any direction.
4. VECTOR DIAGRAMS
Vector diagrams are diagrams, which describe the direction
and relative magnitude of a vector quantity by a vector
arrow.
For example,
the velocity of a car moving down the road vector diagram:
(Vector diagrams describe the velocity of a moving object
during its motion)
5. Multiply vector by a scalar
Multiply the magnitude, direction does not change for
positive scalar, opposite for negative scalar
𝐴
4𝐴
−3𝐴
5. Vector Addition and Subtraction in 1 dimension
A) One dimension vector addition.
𝐴+𝐵 =𝑅
𝑅 is called resultant vector.
head-to-tail method
B) Subtraction:
−𝐵
𝐴 − 𝐵 = 𝐴 + (−𝐵) = 𝑅
finish −𝐵
𝑅
start
𝐴
Follow Commutative Property of Addition
3m
5m
3m
4m
8m
2m
9m
Check page :
http://www.intmath.com/vectors/2vector-addition-1d.php
5m
3m
+4 m
3m
-3 m
+2 m
In one dimension, we can use positive and negative numbers to
represent vector: 𝐴=+4m, 𝐵 = +2m, 𝐶=−3m, then
𝑅=𝐴+𝐵 + 𝐶=4m +2m+(-3m)=3m
Example 2:
+5 + 5=+10
+5 +(- 5)=0
+5 + 10=+15
+5 +(-10) = -5
5 – 15 = -10
10 +(-5) = 5
6. Vectors in 2 dimensional space
A vector in 2 dimensional
space usually is represented
by its magnitude and direction
angle:
vector 𝐴 (2 km, 30o )
Note: the angle is CCW from
positive x-axis, here is from
east direction.
7. Adding Vectors in 2 dimensional
space
head-to-tail method
Example 3
Scale: 1cm=1m
𝑅
𝐴
6.00 m
𝐵
2.00 m
Example 2 continue
R  2.00 m   6.00 m 
2
2
R
2
2.00 m  6.00 m
2
2
 6.32m
𝑅
2.00 m
6.00 m
Adding more than 2 vectors
If displacement vectors A, B, and C
are added together, the result will be
vector R. As shown in the diagram,
vector R can be determined by the
use of an accurately drawn, scaled,
vector addition diagram.
example
Example 4. vector walk
Either using centimeter-sized displacements upon a map or metersized displacements in a large open area, a student makes several
consecutive displacements beginning from a designated starting
position.
𝑅
Draw the resultant
from the tail of the first
vector to the head of
the last vector
Play the simulation in site:
https://phet.colorado.edu/en/simulation/vector-addition