Transcript Two dimensional kinematics

Vectors
• A vector is a quantity that is characterized by
both magnitude and direction.
• Vectors are represented by arrows. The length
of the arrow represents magnitude. The
direction of the arrow represents direction.
• That end of the vector with the arrow point is
called its head. The other and is called its tail.
Vectors
• Vectors do not have position. A vector may be
moved anywhere in a coordinate system, and
still be the same vector as long as it's
magnitude and direction do not change.
• In the next slide all the arrows represent the
same vector.
Figure 3-10 (page 60)
Identical Vectors A at Different Locations
Vectors
• Symbols representing vector quantities have
arrows drawn above them.
A B a b v
• In print vectors are frequently represented by
boldface characters.
Vectors
• Vectors are added either graphically or
mathematically.
• Graphical addition involves placing the tail of
one of vector at the head of another, and
showing the resultant vector by connecting
the tail of the first of vector to the head of the
second.
• The vector arrows must be drawn very
carefully to obtain accurate results.
Figure 3-11a (page 61)
A+B=B+A
Figure 3-11b
A+B=B+A
Vectors
• Multiple vectors may be added into this
manner.
Figure 3-9 (p. 60)
Adding Several Vectors
Vectors
• In order to represent the negative of a vector,
we reverse the head and tail positions.
Figure 3-14a
Vector Subtraction
Vectors
• To add and subtract vectors mathematically,
we must define a coordinate system, and
apply trigonometric methods.
Figure 3-3
A Two-Dimensional Coordinate System
Vectors
• Any vector can be resolved into components.
When the components are added together,
the result is the original vector.
Vectors
• It is convenient to resolve vectors into
components that lie along the axes of the
coordinate system.
Vectors
Vectors
• Standard angle. A standard angle has its
vertex at the origin and is measured from the
positive x axis. If measured counterclockwise
from the positive x axis, the angle is
considered positive. If measured clockwise
from the positive x axis the angle is considered
negative.
Vectors
Standard angle
Vectors
• To resolve a vector into perpendicular
components, we multiply its magnitude by the
appropriate trigonometric function of the
angle between the vector and the horizontal
axis.
Figure 3-7a
Vector Angle
Vectors
• If the angle is between the vector and the
vertical axis, different trigonometric functions
must be used.
Figure 3-7b
Vector Angle
Figure 3-5
A Vector Whose x and y Components Are Positive
Figure 3-6ab
Examples of Vectors with Components of Different Signs
Figure 3-6cd
Examples of Vectors with Components of Different Signs
Vectors
• Unit vectors.
Unit vectors are vectors with a magnitude of 1,
that indicate the direction of a vector. The unit
vectors that indicate the direction along the axes
of a three dimensional cartesian coordinate
system, are
xˆ , yˆ , and zˆ
Vectors
• Unit vectors.
Another frequently used system of notation for unit
vectors is
iˆ, ˆj , kˆ
iˆ
ĵ
k̂
indicates direction along the x axis
indicates direction along the y axis
indicates direction along the z axis
Figure 3-7a
Vector Angle
Vectors
• Unit vectors.
To express the above vector using unit vectors
we write it as follows.
A   A cos   xˆ   A sin   yˆ
The above expression is the vector A in
Cartesian coordinates.
Vectors
Unit vectors
• Adding vectors -to add two or more vectors,
we resolve them using unit vectors as
described above, add the terms multiplied by x̂
together, and add the terms multiplied by ŷ
together. The resultant vector is expressed in
Cartesian coordinates.
Vectors
Unit vectors
• To convert the resultant vector to polar
coordinates (magnitude and angle), we use
the Pythagorean theorem to obtain the
 term multiplied by yˆ 
tan
magnitude, and the


 term multiplied by xˆ 
to obtain the angle θ.
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