Download Chapter 3 – Two Dimensional Motion and Vectors

Chapter 3 – Two Dimensional
Motion and Vectors
3 – 1: Objectives
• Distinguish between a scalar and a vector
• Add and subtract vectors using the
graphical method
• Multiply and Divide Vectors by Scalars
Every physical quantity is either a
scalar or a vector quantity
• Scalar: a physical quantity that can be
completely specified by its magnitude (a
number) with appropriate units.
– Examples: mass, speed, distance and volume
• Vector: a physical quantity that has both
magnitude and direction
– Examples: position, displacement, velocity,
and acceleration
Notation used to represent vector
quantities
• Book uses boldface
type to represent
vector quantities
–
–
–
–
v
a
x
∆x
• Handwritten – place a
“vector symbol” over
the variable
–
–
–
–
v
a
x
∆x
Vectors can be represented by
diagrams
• Arrows are used to
show a vector
quantity that points in
the direction of the
vector.
• The length of the
arrow represents the
magnitude
50 m/s, East
25 m/s, East
Notice, the 50 m/s vector is
twice as long as the 25 m/s
vector
Draw 2 vectors that represent 10 m
east and 15 m west
Notice: The arrow head is pointing in the required direction and the
lengths are drawn to a chosen scale where each unit represents 5 m.
Vector Addition – Graphical Method
• 1. Vectors to be added are physically
placed tip – to – tail (the tip of one vector
touches the tail of the next vector) in any
order
– NOTE: Within a diagram, vectors can be
moved (translated) for the purpose of vector
addition, as long as the direction and the
length remain the same.
Resultant Vector
• the sum of 2 or more vectors
• the solution to a vector addition problem
• also called vector sum
Finding a Resulant Vector
• Found graphically by drawing another
vector that begins at the tail of the first
vector and ends at the tip of the last vector
that is being added.
--NOT TIP-TO-TAIL! Beginning to end.
Graphical Vector Addition in One Dimension
Tip – to - tip
Tail – to - tail
Tip – to tail
NOTES: technically if all vectors are in one – dimension, they would be
drawn on top of each other, these are separated slightly for clarity.
The magnitude of the resultant vector can be found by measuring the
length and converting the number to the proper units using the given scale.
The direction is shown by the arrow tip.
• The diagram shown on the previous page
shows 2 displacement vectors that were
being added (10 m east and 15 m west)
• The resultant vector is obviously 5 m west.
• In one – dimension it is certainly easier to
use the magnitude and a +/- sign for
direction to add the vectors
– Ex. (+10 m) + (-15 m) = -5 m
• The resultant vector is 5 m to the west!
Graphically Add the following 3
displacement vectors (1-dimensional)
• Choose an appropriate scale and draw the graphical
solution to this vector addition problem
225 m north, 175 m south, and 125 m south
Graphical Vector Addition in 2
Dimensions
• The graphical procedure is the same as in 1 –
dimension
– Vectors to be added are physically placed tip – to –
tail (the tip of one vector touches the tail of the next
vector) in any order
– The resultant vector is found graphically by drawing
another vector that begins at the tail of the first vector
and ends at the tip of the last vector
Plus vector 2
Resultant vector
Vector 1
• The magnitude of the resultant vector can
be found by measuring the length and
converting the number to the proper units
using the given scale. (exactly the same
as in 1 – dimension)
• The direction is described differently.
– The direction of a 2 – dimensional vector is
graphically determined with a protractor and is
measured counter-clockwise (CCW) from the
+x - axis
Plus vector 2
Resultant vector
2. Measure the
direction CCW from
the + x - axis
θ
Vector 1
1. Place an x/y
coordinate system at
THE TAIL of your
resultant vector
Important comment!
• If given a vector diagram where the
vectors are not drawn tip - to – tail, you
can move a vector in a diagram so that
you can set up a tip – to – tail situation!
Proceed as before.
Vector 2
Vector 1
Vector 1
Resultant Vector
Vector 2
Hints about vector addition
• When adding vectors:
– 1. The vectors must represent the same
physical quantity (you can’t add velocity and
displacement)
– 2. The vector quantities must have the same
units (you can’t add m and km, you must
convert first)
Resultant Vector
• The resultant vector represents a SINGLE
vector that produces the same RESULT
as the other vectors (addends) acting
together
Example (Displacement)
5m
53º
3m
4m
Walking 3 m east and
then 4 m north puts you
at the same final position
as walking 5 m at an
angle of 53º
Sample problem
140 m
120 m
Find the resultant displacement.
• A person rows due east across the
Delaware River at 8.0 m/s. The current
carries the boat downstream (south) at 2.5
m/s. What is the person’s resultant
velocity?
• Graphical Vector Addition Practice
– Worksheet
– Rulers
– Protractors
Review Problems
1. Two ropes are tied to a tree to be cut down.
The first rope pulls on the tree with a force of
350 N west. The second pulls at 425 N at 320
degrees. What’s the resultant force?
2. A person drives through town 6 blocks north,
then 3 blocks east. They run into a one way
street and have to travel 1 block south to go 2
more blocks east. Finally, the person parks
and walks 2 blocks north to the destination.
What is the person’s displacement?
Part II
Properties of Vectors
• 1. Vectors may be translated in a diagram
(moved parallel to themselves)
• 2. Vectors may be added in any order (Vector
addition is commutative)
• 3. To subtract a vector, add its opposite.
– The opposite of a vector has the same magnitude and
points in the opposite direction. (+/- 180º)
• 4.Multiplying or dividing vectors by scalars
results in vectors
2. Vectors may be added in any
order (Vector addition is
commutative)
3. To subtract a vector, add its
opposite.
A-B
A + (-B)
A
A
-B
B
4.Muliplying or dividing vectors by
scalars results in vectors
A
2A
A/2
Notice: The magnitude is multiplied or divided but the direction
remains the same.
A ball is thrown 25 m at an angle of 30º
Two times this displacement vector is 50 m at an angle to 30º
Sample problems
• Given the following vectors:
A = 50 m South
B = 80 m East
C = 65 m @ 210° D = 110 m @ 140°
Find:
1. A – C
3. ½ B – 4A
2. 3D + B -2A
3-2 Vector Operations
• Objectives:
– Identify appropriate coordinate systems for
solving problems with vectors.
– Apply the Pythagorean Theorem and tangent
function to calculate the magnitude and
direction of a resultant vector.
– Resolve vectors into components using the
sine and cosine functions.
– Add vectors that are not perpendicular
Geometry / Trigonometry Review
• Pythagorean Theorem – The square of
the hypotenuse of a right triangle is equal
to the sum of the squares of its legs
• c2 = a2 + b2
c
a
b
• Trigonometric Ratios:
B
Hypotenuse (c)
Leg (a)
C
A
Leg (b)
Opposite side Cos θ = Adjacent side
Sin θ =
hypotenuse
hypotenuse
Tan θ =
Opposite side
Adjacent side
Using Trig. Ratios
• Given an acute angle of a right triangle, to
find the ratio of 2 specific sides of the
triangle, enter the appropriate function
(sine, cosine, tangent) of the angle in your
calculator.
Sin(20º)=b/c
c
Cos(20º) = a/c
b
20º
a
Tan(20º) = b/a
• To find an acute angle of a right triangle,
enter the inverse of the appropriate
function of the ratio of the 2 corresponding
sides.
c
θ
θ = sin-1(a/c)
θ = cos-1(b/c)
b
θ = tan-1(a/b)
a
• Trigonometry Review Practice Worksheet
Part III
Vector Addition – Analytical Method
• Case #1 (easiest method):
– Adding 2 Vectors that are perpendicular
Resultant, R
B
θ
R = magnitude of the
resultant vector
R=
A
A2 + B2
The angle, θ, of the triangle can be
found using the tan-1 function and
THEN CONVERT it to the direction
measured CCW from the +x - axis
Example for Case #1
• Add the following 2 velocity vectors.
5 m/s west (180º) and 8 m/s north (90º)
R
8 m/s
R2 = 52 + 82
R = 9.4 m/s
θ
θ = tan-1 (8/5)
θ = 58º
5 m/s
The direction (measured CCW
from the +x – axis) is found by
subtracting 180 – 58 = 122º
R = 9.4 m/s <122º
Case #2: Adding more than 2
perpendicular vectors
• First, find the vector sum of all of the
horizontal vectors, call this Rx.
• Second, find the vector sum of all of the
vertical vectors, call this Ry.
• Find the vector sum of Rx and Ry
– By following the method from Case #1
Example of Case #2
• A boyscout walks 8 m east, 2 m north, 6 m
east, 10 m south, 3 m east, 5 m south and
3 m west.
Horizontal Vectors
Vertical Vectors
+ 8m
+ 2m
+ 6m
- 10m
+ 3m
- 5m
- 3m
+14 m
-13 m
Rx
R2 = 142 + 132
θ
Ry
R
R = 19.1 m <317º
R = 19.1 m
θ = tan-1 (13/14)
θ = 43º
The direction
(measured CCW
from the +x – axis)
is found by
subtracting 360 – 43
= 317º
Vector Resolution (opposite
process of adding 2 vectors)
• Any vector acting at an angle can be
replaced with 2 vectors that act
perpendicular to each other, one
horizontal and one vertical. (The 2 vectors
working together are equivalent to the
single vector acting at an angle.)
Step 1
• Sketch the given vector with the tail
located at the origin of an x-y coordinate
system. (Ex. 25 m at an angle of 36º)
25 m
36º
Step 2
• Draw a line segment from the tip of the
vector perpendicular to the x-axis
25 m
36º
Notice, you now have a right
triangle with a known
hypotenuse and known angle
measurements
Step 3
• Replace the perpendicular sides of the
right triangle with vectors drawn tip – to tail
25 m
Step 4
• Use sine and cosine functions to find the
horizontal and vertical components of the
given vector.
25 m
36º
Cos(36) = Rx/25
Rx = 25cos(36)
Rx = 20.2 m
sin(36) = Ry/25
Ry = 25sin(36)
Ry = 14.7 m
Rx
Ry
Important
• Remember that the 2 components acting
together gives the same result as the
single vector acting at an angle.
• ****The 2 components can be used to
REPLACE the single vector****
Example #2
• Find the components of 16m at 200º
200º
20º
You have 2 choices at this point.
You can use the directional angle of 200 and not worry
about the sign of the components (the calculator will do it for you).
OR, you can use 20 and YOU must remember to put –
signs when the component points down or to the left
Example #2
• Find the components of 16m at 200º
200º
20º
Rx = 16cos(200) = -15 m
Ry = 16sin(200) = -5.5 m
Rx = -16cos(20) = -15 m
Ry = -16sin(20) = -5.5 m
Case #3 – Adding Vectors at
Angles (not perpendicular)
• When vectors to be added are not
perpendicular, they do not form sides of a
right triangle.
Look at the geometry for the
situation
Resultant Vector
Ry2
Ry1 + Ry2
Rx2
Rx1
Rx1 + Rx2
Ry1
• Notice, the length of the horizontal
component of the resultant vector is equal
to the sum of the lengths of the horizontal
components of the vectors that are being
added together.
• This is also true for the vertical
component.
Steps for solving Case #3
Problems
• 1. Resolve each vector that is being added
(addends) into components.
• 2. Add all the horizontal components
together and all the vertical components
together (Case #2)
• 3. Use the Pythagorean Theorem and trig
ratios to find the resultant vector (Case #1)
Example for Case #3
• Add these 2 vectors together: 10 m/s at 0º
and 12 m/s at 25º (Find the
resultant vector, R at θ)
R
12 m/s
θ
10 m/s
25º
Example for Case #3
Find components of each vector
x
R
12 m/s
θ
10 m/s
25º
y
Vector 1
10cos(0)
10sin(0)
Vector 2
12cos(25) 12sin(25)
Example for Case #3
Add horizontal and vertical
components
x
R
12 m/s
θ
Vector 1
10cos(0)
Vector 2
12cos(25) 12sin(25)
10sin(0)
25º
21
10 m/s
y
5.1
Example for Case #3
Find the magnitude of the
resultant vector using the
Pythagorean Theorem
R2 = 212 + 5.12
R
5.1
θ
R = 21.6 m/s
θ = tan-1 (5.1/21)
θ = 14º
21
R = 21. m/s at 14º