Physics 1710 Chapter 3 Vectors

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• Demonstration
• Egg Toss

REVIEW
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• Why did the egg not break the first time it was
  caught but did the second time?

REVIEW
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• Different acceleration

a (vfinal 2 v initial 2)/ (2?x)
Why wear a seat belt or use air bags?
REVIEW
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• Seat belt
• Air Bag Video

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• 1' Lecture
• A Vector is a quantity that requires two or
  more numbers to define it and acts like the
  displacement vector.
• The magnitude of a vector is the square root of
  the sum of the squares of its components.
• A vector makes an angle to the x-axis whose
  tangent is equal to the ratio of the y-component
  to the x-component.

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• Is it far to Budapest?

Stranded Motorist asks horse cart driver, Is it
far to Budapest? Nem! It is not far.
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Physics 1710 Chapter 3 Vectors
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Then, may I have a ride? Egan! Climb
up. After a long time the Motorist says, I
thought you said it was not far.

What is the problem?
The difference between distance and displacement.
The driver replies, Oh! Now it is very far to
Budapest.
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• Where is the Student Union?

Turn to your classmate and the one in the odd
numbered seat, tell the other where is the
Student Union.
Position is a vector.
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• A Scalar is a entity that requires only one
  number to characterize it fully. (Like a scale.)
• Examples
• What time is it?
• What is your weight?
• What is the temperature of the room?

What is the weight of 100. Kg man? Weight g m
9.80 N/kg (100. kg) 980 N.
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• A vector is a quantity that requires more than
  one component to tell the whole story.
• Example
• Where is the treasure buried in the field?

Use orthogonal, that is, perpendicular axes.
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• Location in Manhattan

(4,2)
(2,4)
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• Position in 2-Dimensions or higher is a VECTOR.
  We use boldface, not italic, to denote a vector
  quantity, italics to denote the scalar
  components.
• We often represent a vector as a position on a
  graph with an arrow connecting the origin to the
  position.

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Physics 1710Chapter 2 Motion in One
DimensionII
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• 2-Dimensional Vector

Position Vector r r (x,y) x i y j x r cos
? y r sin ? r r v(x 2 y 2), ? tan
1(y/x)
Y(m)
x
r
y
X (m)
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80/20 Fact

• The length of the arrow represents the magnitude
  of the vector. In orthogonal coordinates, the
  magnitude of vector A given by
• A v Ax2 Ay2 Az2

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80/20 Fact

• The direction of the vector A is characterized
  (two dimensions) by the angle ? it makes with
  the x-axis.
• tan ? Ay / Ax

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Physics 1710Chapter 2 Motion in One
DimensionII
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• 2-Dimensional Vector

Position Vector r r r v(x 2 y 2)
v(2.0 2 1.5 2) v(4.0 2.25 ) v(6.25) 2.5
m
Y(m)
x
r
y
X (m)
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80/20 Fact

• One may combine vectors by vector addition
• C A B
• Then
• C x Ax Bx Cy Ay By
• Key point
• Add the components separately.
• Observe strict segregation of x and y parts.

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80/20 Fact

• The product of a scalar and a vector is a vector
  for which every component is multiplied by the
  scalar
• C k A
• Cx k Ax
• Cy k Ay
• Cz k Az

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• N.B. ( Note Well)
• ?A B? ? (A B)

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• Note
• ?A B? v(Ax Bx ) 2 (Ay By ) 2 (A
  B)
• Proof
• (Ax Bx ) 2 (Ay By ) 2 (AB)2 A2 2AB B2
  
• LHS Ax 2 Ay2 Bx 2 By 2 2AxBx 2AyBy
• RHS Ax2 Ay2 Bx 2 By 2 2v(Ax2 Bx2 Ay2
  By2 Ay2 Bx2 Ax2 By2)
• LHS RHS
• 2AxBx 2AyBy 2v(Ax2 Bx2 Ay2 By2 Ay2 Bx2
  Ax2 By2)
• Ax 2Bx 2 2 AxBx AyBy Ay2 By 2 Ax2 Bx2
  Ay2 By2 Ay2 Bx2 Ax2 By2
• 2 AxBx AyBy Ay2 Bx2 Ax2 By2
• iff 0 (AyBx - Ax By) 2 ?

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80/20 Fact

• We often designate the components of the vector
  by unit vectors ( i, j, k ) the x,y, and z
  components, respectively.
• Thus, 2.0 i 3.0 j has an x-component of 2.0
  units and a y-component of 3.0 units.
• Or (2.0, 3.0)

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• Summary
• To add vectors, simply add the components
  separately.
• Use the Pythagorean theorem for the magnitude.
• Use trigonometry to get the angle.
• The vector sum will always be equal or less than
  the arithmetic sum of the magnitudes of the
  vectors.

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Physics 1710Chapter 2 Motion in One
DimensionII
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• 1' EssayOne of the following

• The main point of todays lecture.
• A realization I had today.
• A question I have.

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