Chapter 3 – Two Dimensional Motion and Vectors

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Chapter 3 Two Dimensional Motion and Vectors
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3 1 Objectives

• Distinguish between a scalar and a vector
• Add and subtract vectors using the graphical
  method
• Multiply and Divide Vectors by Scalars

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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

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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

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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

Notice, the 50 m/s vector is twice as long as the
25 m/s vector
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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.
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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.

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Resultant Vector

• the sum of 2 or more vectors
• the solution to a vector addition problem
• also called vector sum

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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.

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Graphical Vector Addition in One - Dimension
Tip to - tip
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.
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• 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!

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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

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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

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• 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

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?
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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.

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Vector 2
Vector 1
Vector 1
Resultant Vector
Vector 2
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Hints about vector addition

• When adding vectors
• 1. The vectors must represent the same physical
  quantity (you cant add velocity and
  displacement)
• 2. The vector quantities must have the same units
  (you cant add m and km, you must convert first)

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Resultant Vector

• The resultant vector represents a SINGLE vector
  that produces the same RESULT as the other
  vectors (addends) acting together

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Example (Displacement)
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
4 m
5 m
53º
3 m
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Sample problem

• 140 m
• 120 m
• Find the resultant displacement.
  

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• 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
  persons resultant velocity?

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• Graphical Vector Addition Practice
• Worksheet
• Rulers
• Protractors

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Review Problems

• 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. Whats the resultant force?
• 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
  persons displacement?

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Part II
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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

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2. Vectors may be added in any order (Vector
addition is commutative)
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3. To subtract a vector, add its opposite.
A - B
A (-B)
A
A
-B
B
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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
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Sample problems

• Given the following vectors
• A 50 m South B 80 m East
• C 65 m _at_ 210 D 110 m _at_ 140
• Find
• A C 2. 3D B -2A
• 3. ½ B 4A

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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

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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
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• Trigonometric Ratios

Adjacent side
Opposite side
Opposite side
Cos ?
Tan ?
Sin ?
hypotenuse
Adjacent side
hypotenuse
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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 Cos(20º) a/c Tan(20º) b/a
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• To find an acute angle of a right triangle, enter
  the inverse of the appropriate function of the
  ratio of the 2 corresponding sides.

? sin-1(a/c) ? cos-1(b/c) ? tan-1(a/b)
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• Trigonometry Review Practice Worksheet

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Part III
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Vector Addition Analytical Method

• Case 1 (easiest method)
• Adding 2 Vectors that are perpendicular

R magnitude of the resultant vector R A2
B2
The angle, ?, of the triangle can be found using
the tan-1 function and THEN CONVERT it to the
direction measured CCW from the x - axis
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Example for Case 1

• Add the following 2 velocity vectors.
• 5 m/s west (180º) and 8 m/s north (90º)

R2 52 82 R 9.4 m/s ? tan-1 (8/5) ?
58º The direction (measured CCW from the x
axis) is found by subtracting 180 58 122º
R
8 m/s
?
5 m/s
R 9.4 m/s lt122º
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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

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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.

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Horizontal Vectors
Vertical Vectors
8m 6m 3m - 3m

• 2m
• 10m
• 5m

-13 m
14 m
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R2 142 132 R 19.1 m ? tan-1 (13/14) ?
43º The direction (measured CCW from the x
axis) is found by subtracting 360 43 317º
R 19.1 m lt317º
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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.)

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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)

36º
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Step 2

• Draw a line segment from the tip of the vector
  perpendicular to the x-axis

36º
Notice, you now have a right triangle with a
known hypotenuse and known angle measurements
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Step 3

• Replace the perpendicular sides of the right
  triangle with vectors drawn tip to - tail

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Step 4

• Use sine and cosine functions to find the
  horizontal and vertical components of the given
  vector.

Ry
36º
Cos(36) Rx/25 Rx 25cos(36) Rx 20.2 m
sin(36) Ry/25 Ry 25sin(36) Ry 14.7 m
Rx
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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

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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
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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
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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.

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Look at the geometry for the situation
Resultant Vector
Ry1 Ry2
Rx1 Rx2
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• 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.

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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)

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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 ?)

R
12 m/s
25º
?
10 m/s
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Example for Case 3

• Find components of each vector

x
y
Vector 1
10cos(0)
10sin(0)
R
Vector 2
12cos(25)
12sin(25)
12 m/s
25º
?
10 m/s
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Example for Case 3

• Add horizontal and vertical components

x
y
Vector 1
10cos(0)
10sin(0)
R
Vector 2
12cos(25)
12sin(25)
12 m/s
25º
?
10 m/s
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Example for Case 3

• Find the magnitude of the resultant vector using
  the Pythagorean Theorem

R2 212 5.12
R
R 21.6 m/s ? tan-1 (5.1/21) ? 14º
5.1
?
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R 21. m/s at 14º