Post on 28-Dec-2015
Adding Vectors, Rules
When two vectors are added, the sum is independent of the order of the addition. This is the Commutative
Law of Addition A B B A
Adding Vectors, Rules cont.
When adding three or more vectors, their sum is independent of the way in which the individual vectors are grouped This is called the Associative Property of Addition
A B C A B C
Adding Vectors, Rules final
When adding vectors, all of the vectors must have the same units
All of the vectors must be of the same type of quantity For example, you cannot add a displacement to a
velocity
Negative of a Vector
The negative of a vector is defined as the vector that, when added to the original vector, gives a resultant of zero Represented as
The negative of the vector will have the same magnitude, but point in the opposite direction
A
0 A A
Subtracting Vectors
Special case of vector addition
If , then use Continue with standard
vector addition procedure
A B
A B
Subtracting Vectors, Method 2
Another way to look at subtraction is to find the vector that, added to the second vector gives you the first vector
As shown, the resultant
vector points from the tip of the second to the tip of the first
A B C
Multiplying or Dividing a Vector by a Scalar
The result of the multiplication or division of a vector by a scalar is a vector
The magnitude of the vector is multiplied or divided by the scalar
If the scalar is positive, the direction of the result is the same as of the original vector
If the scalar is negative, the direction of the result is opposite that of the original vector
Component Method of Adding Vectors
Graphical addition is not recommended when High accuracy is required If you have a three-dimensional problem
Component method is an alternative method It uses projections of vectors along coordinate
axes
Components of a Vector, Introduction
A component is a projection of a vector along an axis Any vector can be
completely described by its components
It is useful to use rectangular components These are the projections
of the vector along the x- and y-axes
Vector Component Terminology
are the component vectors of They are vectors and follow all the rules for
vectors
Ax and Ay are scalars, and will be referred to as the components of
x yandA A
A
A
Components of a Vector
Assume you are given a vector
It can be expressed in terms of two other vectors, and
These three vectors form a right triangle
A
xA
yA
x y A A A
Components of a Vector, 2
The y-component is moved to the end of the x-component
This is due to the fact that any vector can be moved parallel to itself without being affected This completes the
triangle
Components of a Vector, 3
The x-component of a vector is the projection along the x-axis
The y-component of a vector is the projection along the y-axis
This assumes the angle θ is measured with respect to the x-axis If not, do not use these equations, use the sides of the
triangle directly
cosxA A
sinyA A
Components of a Vector, 4
The components are the legs of the right triangle whose hypotenuse is the length of A
May still have to find θ with respect to the positive x-axis
2 2 1and tan yx y
x
AA A A
A
Components of a Vector, final
The components can be positive or negative and will have the same units as the original vector
The signs of the components will depend on the angle
Unit Vectors
A unit vector is a dimensionless vector with a magnitude of exactly 1.
Unit vectors are used to specify a direction and have no other physical significance
Unit Vectors, cont.
The symbols
represent unit vectors They form a set of
mutually perpendicular vectors in a right-handed coordinate system
Remember,
kand,j,i
ˆ ˆ ˆ 1 i j k
Viewing a Vector and Its Projections
Rotate the axes for various views
Study the projection of a vector on various planes x, y x, z y, z
Unit Vectors in Vector Notation
Ax is the same as Ax
and Ay is the same as Ay etc.
The complete vector can be expressed as
i
j
ˆ ˆ
x yA AA i j
Adding Vectors Using Unit Vectors
Using Then
and so Rx = Ax + Bx and Ry = Ay + By
ˆ ˆ ˆ ˆ
ˆ ˆ
ˆ ˆ
x y x y
x x y y
x y
A A B B
A B A B
R R
R i j i j
R i j
R i j
2 2 1tan yx y
x
RR R R
R
R A B
Adding Vectors with Unit Vectors
Note the relationships among the components of the resultant and the components of the original vectors
Rx = Ax + Bx
Ry = Ay + By
Three-Dimensional Extension
Using Then
and so Rx= Ax+Bx, Ry= Ay+By, and Rz =Ax+Bz
ˆ ˆ ˆ ˆ ˆ ˆ
ˆ ˆ ˆ
ˆ ˆ ˆ
x y z x y z
x x y y z z
x y z
A A A B B B
A B A B A B
R R R
R i j k i j k
R i j k
R i j k
2 2 2 1cos , .xx y z
RR R R R etc
R
R A B
Example 3.5 – Taking a Hike
A hiker begins a trip by first walking 25.0 km southeast from her car. She stops and sets up her tent for the night. On the second day, she walks 40.0 km in a direction 60.0° north of east, at which point she discovers a forest ranger’s tower.
Example 3.5 (A) Determine the components
of the hiker’s displacement for each day.
A
Solution: We conceptualize the problem by drawing a sketch as in the figure above. If we denote the displacement vectors on the first and second days by and respectively, and use the car as the origin of coordinates, we obtain the vectors shown in the figure. Drawing the resultant , we can now categorize this problem as an addition of two vectors.
B A
R
Example 3.5 We will analyze this
problem by using our new knowledge of vector components. Displacement has a magnitude of 25.0 km and is directed 45.0° below the positive x axis.
From Equations 3.8 and 3.9, its components are:
cos( 45.0 ) (25.0 km)(0.707) = 17.7 km
sin( 45.0 ) (25.0 km)( 0.707) 17.7 kmx
y
A A
A A
The negative value of Ay indicates that the hiker walks in the negative y direction on the first day. The signs of Ax and Ay also are evident from the figure above.
A
Example 3.5
The second displacement has a magnitude of 40.0 km and is 60.0° north of east.
Its components are:
cos60.0 (40.0 km)(0.500) = 20.0 km
sin 60.0 (40.0 km)(0.866) 34.6 kmx
y
B B
B B
B
Example 3.5
(B) Determine the components of the hiker’s resultant displacement for the trip. Find an expression for in terms of unit vectors.
Solution: The resultant displacement for the trip has components given by Equation 3.15:
Rx = Ax + Bx = 17.7 km + 20.0 km = 37.7 km
Ry = Ay + By = -17.7 km + 34.6 km = 16.9 km
In unit-vector form, we can write the total displacement as
R
ˆ ˆR = (37.7 + 16.9 ) kmi j
R A B
R
R
Example 3.5
Using Equations 3.16 and 3.17, we find that the resultant vector has a magnitude of 41.3 km and is directed 24.1° north of east.
R
Let us finalize. The units of are km, which is reasonable for a displacement. Looking at the graphical representation in the figure above, we estimate that the final position of the hiker is at about (38 km, 17 km) which is consistent with the components of in our final result. Also, both components of are positive, putting the final position in the first quadrant of the coordinate system, which is also consistent with the figure.
R
R
R
Motion in Two Dimensions
Using + or – signs is not always sufficient to fully describe motion in more than one dimension Vectors can be used to more fully describe motion Will look at vector nature of quantities in more detail
Still interested in displacement, velocity, and acceleration
Will serve as the basis of multiple types of motion in future chapters
Position and Displacement
The position of an object is described by its position vector,
The displacement of the object is defined as the change in its position
r
f ir r r
General Motion Ideas
In two- or three-dimensional kinematics, everything is the same as as in one-dimensional motion except that we must now use full vector notation Positive and negative signs are no longer
sufficient to determine the direction
Average Velocity
The average velocity is the ratio of the displacement to the time interval for the displacement
The direction of the average velocity is the direction of the displacement vector
The average velocity between points is independent of the path taken This is because it is dependent on the displacement, also
independent of the path
avg t
rv
Instantaneous Velocity
The instantaneous velocity is the limit of the average velocity as Δt approaches zero
As the time interval becomes smaller, the direction of the displacement approaches that of the line tangent to the curve
0lim
t
d
t dt
r rv
Instantaneous Velocity, cont
The direction of the instantaneous velocity vector at any point in a particle’s path is along a line tangent to the path at that point and in the direction of motion
The magnitude of the instantaneous velocity vector is the speed The speed is a scalar quantity
Average Acceleration
The average acceleration of a particle as it moves is defined as the change in the instantaneous velocity vector divided by the time interval during which that change occurs.
f i
avgf it t t
v v va
Average Acceleration, cont
As a particle moves, the direction of the change in velocity is found by vector subtraction
The average acceleration is a vector quantity directed along
f iv v v
v
Instantaneous Acceleration
The instantaneous acceleration is the limiting value of the ratio as Δt approaches zero
The instantaneous equals the derivative of the velocity vector with respect to time
0lim
t
d
t dt
v va
tv
Producing An Acceleration
Various changes in a particle’s motion may produce an acceleration The magnitude of the velocity vector may change The direction of the velocity vector may change
Even if the magnitude remains constant Both may change simultaneously
Kinematic Equations for Two-Dimensional Motion
When the two-dimensional motion has a constant acceleration, a series of equations can be developed that describe the motion
These equations will be similar to those of one-dimensional kinematics
Motion in two dimensions can be modeled as two independent motions in each of the two perpendicular directions associated with the x and y axes Any influence in the y direction does not affect the motion
in the x direction
Kinematic Equations, 2
Position vector for a particle moving in the xy plane
The velocity vector can be found from the position vector
Since acceleration is constant, we can also find an expression for the velocity as a function of time:
ˆ ˆx yr i j
ˆ ˆx y
dv v
dt
rv i j
f i tv v a
Kinematic Equations, 3
The position vector can also be expressed as a function of time: This indicates that the position vector is the sum
of three other vectors: The initial position vector The displacement resulting from the initial velocity The displacement resulting from the acceleration
21
2f i it tr r v a
Kinematic Equations, Graphical Representation of Final Velocity
The velocity vector can be represented by its components
is generally not along the direction of either or
fv
iva
Kinematic Equations, Graphical Representation of Final Position
The vector representation of the position vector
is generally not along the same direction as or as
and are generally not in the same direction
fr
iva
fv
fr
Graphical Representation Summary
Various starting positions and initial velocities can be chosen
Note the relationships between changes made in either the position or velocity and the resulting effect on the other