Vectors

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Vectors

• Measurements With Directions!

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Any measurement with a direction is a vector.

• 5.6 km North
• 4.2m/s Forward
• 9.8m/s2 Downward
• 5.8x102N To the right
• Measurements without directions are known as
  scalars.

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The Resultant

• The result of taking into account more than one
  vector is known as the resultant or resulting
  vector.

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If vectors are linear (along the same line)

• Add if they are in the same direction
• Subtract any vectors that are in the opposite
  direction

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Examples

• A man walks 6.0m East, then 2.0m West.

• Two women push the same direction (E) on a box
  with forces of 40N and 60N.

R 6.0m (E) 2.0m (W) 4.0m (E)
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Directions
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Vector Diagrams

• Procedure for drawing vector diagrams
• Start from a point of origination.
• Draw the first vector to scale as an arrow
  pointing in its direction.
• Draw any additional vectors (to scale) from the
  end of the previous vector.
• The resultant is the distance from the origin to
  the end of the vectors
• Resultant direction is from the origin pointing
  to the end point.

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Vector Diagram 1

• Start from a point of origination.

2. Draw the first vector to scale as an arrow
pointing in its direction.
3. Draw any additional vectors (to scale) from
the end of the previous vector.
R NE
4. The resultant is the distance from the
origin to the end of the vectors
5. Resultant direction is from the origin
pointing to the end point.
40N N 60N E
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Vector Diagram 2
40m N 60m E 20m S 25m W

• Start from a point of origination.

2. Draw the first vector to scale as an arrow
pointing in its direction.
3. Draw any additional vectors from the end of
the previous vector.
4. The resultant is the distance from the
origin to the end of the vectors
R NE
5. Resultant direction is from the origin
pointing to the end point.
20 m N 35 m E 40 m NE
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If vectors are at a right angle (90)
Use the Pythagorean theorem (yes, from geometry!)

• A person walks 4.0m East then 3.0m South

R 5m ? 5.0 m SE
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Calculate the resulting Force
R ?
R 10N ? 1.0x101 N NE
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Problem 1

• A man drives 12km South then 9.0km West.
• What total distance did he travel?
• What is his resulting displacement (amount and
  direction)?

• Distance 12km 9.0km 21Km

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Problem 2

• An evil monkey climbs 3.0m up a ladder, then 5.0m
  across a scaffolding.
• What total distance did the monkey travel?
• What amount is the monkey displaced?

a) distance 3.0m 5.0m 8.0m
b) Displacement R
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Problem 3

• A person puts 62N of force on a 34kg box in an
  Eastward direction while another person puts a
  42N force on the box in a Northward direction.
• What is the resultant force (amount and
  direction)?
• What amount is the box accelerated?

b) a Fnet/m 75N/34kg 2.205m/s2 ? 2.2 m/s2
NE
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Lab 7 Shopping in Baltimore

• Compare walking distance and displacement as you
  travel to 5 locations around downtown.
• Calculate real distance and displacement vectors
  using map scale.
• 0.4 km 400 m
• On the map 400 m
• 3.0 cm, so 1 cm
• 1 cm 130 m
• This is your scale conversion factor

Click Here to Download Map
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Example

• Draw 2 orthogonal vectors to get from Lombard
  Calvert to Fayette Gay

Lombard to Fayette ? (2.3 cm) ?(2.3 cm)(130
m/cm) 300 m N
Calvert to Gay ? 2.7 cm ?(2.7 cm)(130 m/cm) 350
m E
Total distance 300 m 350 m 650 m
Resultant Displacement R v(a2 b2) v(3002
3502) 212 500 460 m NE
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Check Your Work

• Measure the (resultant) displacement in cm and
  calculate error for each trip

Ex Lombard Calvert to Fayette Gay Length of
resultant  3.6 cm Displacement 3.6 x 130 470
m
error 100(A-P)/A 100(470 460) / 460
2.2
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For each of the following trips

• Trip 1 Federal Hill Park (SW corner) to corner
  of Pratt St. Light St.
• Trip 2 Corner of Pratt St. Light St. to
  Hopkins M.C. (Bway Orleans)
• Trip 3 Bway Orleans to corner of Charles St.
  Chase St.
• Trip 4 Corner of Charles St. Chase St. to
  foot of Broadway
• Trip 5 Foot of Broadway to water end of
  Lancaster
• Measure the orthogonal distances (ie. N, S, E, or
  W) in cm to the nearest tenth.
• Convert measurements to m. (use scale 1.0cm
  130m)
• Calculate total distance travelled.
• Calculate displacement vector. (show work,
  include direction)
• When finished, check resultant by measuring the
  final displacement in cm, converting to m
  (include direction)
• Calculate error using orthogonal vector
  resultant as ACTUAL