Vectors and Two Dimensional Motion

1
Chapter 3

• Vectors and Two Dimensional Motion

2
Vectors

• An arrow can be used to represent the magnitude
  and direction of a vector quantity.
• The length of the arrow, drawn to scale,
  indicates the magnitude of the vector quantity.
  The direction of the arrow indicates the
  direction of the vector quantity. Since velocity
  is a vector quantity, we can represent velocity
  with an arrow. The arrow is called a vector.

3
Properties of Vectors

• Adding Vectors

B
A
Resultant Vector R A B
4
Properties of Vectors

• Negative of a Vector

A
A
5
Properties of Vectors

• Subtracting Vectors
• A B A (B)

B
B
R A ( B)
A
6
Properties of Vectors

• Multiplication of Vectors (Multiplying a vector
  by a scalar)

A
A
A
A
3A
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Do Now 1/5/09

• 1.) Vector A has a magnitude of 5 cm in length
  and points along the positive x axis. Vector B
  has a magnitude of 3 cm in length and points
  along the negative y axis. Find (a) the vector
  sum A B and (b) the vector difference A B.
• 2.) Vector C has a magnitude of 7 cm in length
  and points east. Vector D has a magnitude of 4
  cm and points south. Find the resultant vector R
  C D.

8
Graphical Method Drawing Vectors to Scale

• A man headed east drives his car with a velocity
  of 40 m/s.
• Use a scale drawing of the velocity vector.
• Let 1 cm 10 m/s

4 cm 40 m/s
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Drawing vectors to scale

• A hiker walks 40 km in a direction 30 west of
  north.
• Scale 1 cm 10 km

30
10
N
EAST OF NORTH
WEST OF NORTH
NORTH OF WEST
NORTH OF EAST
W
E
SOUTH OF EAST
SOUTH OF WEST
EAST OF SOUTH
WEST OF SOUTH
S
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• Example 3.1 p. 59
• 1.) An airplane flies 200 km due west from city A
  to city B and then 300 km in the direction of 30
  north of west from city B to city C. How far is
  city C from city A?
• 2.) A dog searching for a bone walks 3.5 m south,
  then 7 m at an angle of 45 north of east, and
  finally 14 m west. Find the dogs resultant
  displacement vector, using graphical techniques.

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• 3.) A roller coaster moves 100 m westward, then
  travels 75 m at an angle of 40 south of east.
  What is its displacement from its starting point?
• 4.) A novice pilot sets a planes controls,
  thinking the plane will fly at 250 km/h to the
  north. If the wind blows at 100 km/h toward the
  southwest, what is the planes resultant velocity?

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Do Now 1/6/09

• 1.) A hiker walks 200 km west, then walks 150 km
  in a direction 60 south of west. Find the
  magnitude and direction of his resultant
  displacement.
•  
• 2.) A plane flies at 70 km/h eastward. It
  encounters a 50 km/h crosswind that causes it to
  fly northward. Find the magnitude and direction
  of the planes resultant displacement.

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Do Now 1/7/09

• 1.) A roller coaster moves 100 m westward, then
  travels 75 m at an angle of 35 North of West.
  What is its displacement from its starting point?
• 2.) An airplane flies at 250 km/h to the north.
  It turns and flies 300 km/h at a direction 60
  East of North. Find the magnitude and direction
  of the planes resultant velocity.

15
Components of a Vector
A
y component
Ay
?
x component
Ax
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Components of a vector

• Any vector A can be expressed as the sum of two
  vectors, Ax and Ay.
• Therefore, A Ax Ay where Ax and Ay are the
  component vectors of A
• 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

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Components of a vector

• The components are the legs of the right triangle
  whose hypotenuse is A
• A plane flies at 70 km/h eastward. It encounters
  a 50 km/h crosswind that causes it to fly
  northward. Using the Pythagorean Theorem and
  tangent, find the magnitude and direction of the
  planes resultant displacement.

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• An airplane flies 120 m/s west when it encounters
  a crosswind blowing south at 60 m/s. Using the
  Pythagorean Theorem and tangent, find the
  magnitude and direction of the airplanes
  resultant velocity.

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Do Now 1/8/09

• 1.) A hiker walks 4 miles west. He turns and
  walks 5 miles south. Find the magnitude and
  direction of the hikers resultant displacement
  using the Pythagorean Theorem and tangent
  (Algebraic method).
• 2.) Find the x and y components for the following
  vectors
• (a) (b)

35
6.5 m
65
55 km
20
B
By
Bx
A
Ay
Ax
21
Rx Ax Bx Ry Ay By
B
R
Ry
A
Rx
22

• Example 3.3 p. 62
• 1.) A hiker walks 75 m northwest on the first day
  of her hiking trip. On the second day, she walks
  100 m in a direction 40 north of east. Find the
  magnitude and direction of her resultant
  displacement using the algebraic method.

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Do Now 1/7/09

• A golfer takes two putts to get his ball into the
  hole once he is on the green. The first putt
  displaces the ball 6.00m east, and the second,
  5.40 m south. What displacement would have been
  needed to get the ball into the hole on the first
  putt? (Use algebra!)

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Do Now 1/9/09

• A plane flies from base camp to lake A, a
  distance of 200 km at a direction 20 north of
  east. After dropping off supplies it flies to
  lake B, which is 150 km and 30 west of north
  from lake A. Algebraically determine the
  magnitude and direction of the resultant
  displacement of the plane.

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• The horizontal component of the projectiles
  velocity remains constant. The vertical
  component changes throughout its trajectory
  (path).
• Since horizontal velocity is constant, there is
  no horizontal acceleration. Vertical
  acceleration will always be -9.8 m/s2 (the
  negative sign indicates that positive y for the
  vertical motion is assumed to be upward.)
• When a projectile is shot at an angle, the
  vertical velocity at the top of its path is
  zero, leaving only the horizontal component of
  velocity. Therefore, the speed at the top is
  equal to the horizontal component of the
  projectiles velocity at any point. The minimum
  speed of the projectile occurs at the peak of its
  trajectory.
• When air resistance is negligible, the projectile
  will hit the ground with the same speed it had
  originally when it was projected upward. The
  speed that the projectile loses going up is
  therefore the same speed it gains coming down.

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Projectile Motion

• ?x vx0t (v0cos ?0)t
• gives us the horizontal displacement of the
  projectile.
• The following equations are to be used for motion
  in the vertical (y) direction vy0 denotes
  initial velocity in the y direction and g is the
  free fall acceleration the negative sign
  indicates that positive y for the vertical motion
  is assumed to be upward.
• vy vy0 gt
• ?y vy0t 1/2gt2
• vy2 vy02 2g?y

p. 75 22
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Do Now 1/13

• (No notes or text may be used)
• Sketch the parabolic path of a projectile that
  leaves the origin with a velocity of v0.
• (a)What is the vertical (y) component of the
  acceleration at any point in the projectiles
  path?
• (b)What can you say about the horizontal (x)
  component of the velocity at any point in the
  projectiles path? What does this tell you about
  the horizontal component of the acceleration?
• (c) What is the vertical velocity at the peak of
  the trajectory?
• (d) What is the horizontal velocity at the peak
  of the trajectory?