Vectors
M Sittig
AP Physics B
Summer Course 2012
2012年AP物理B暑假班
Scalars
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Examples:
3 years old
10 kg
0.33 J
3.0 × 108 m/s
Scalars
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Scalar: a physical quantity with a magnitude
but no direction.
Example of scalar quantities: time, mass,
speed, energy, work, and distance.
Example of scalar
measurement:
Vector Basics
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Examples:
70 km/hr, South
9.8 m/s2 at 145°
6 meters down
Vector Basics
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Vector: a physical quantity with a magnitude
and a direction.
Examples of vector quantities: velocity,
acceleration, displacement, force, fields.
Example of scalar
measurement:
What is this?
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Magnitudes?
Directions?
A=3 m at 60°
B=3 m at 180°-β
Or 3 m at β°
above the
negative x-axis
C=1.5 m at 270°
Or…
Adding and Subtracting Vectors
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What is the
maximum
magnitude of
the sum of
these vectors?
The minimum?
Adding and Subtracting Vectors
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Geometrically: Vectors as arrows, lined up
head-to-tail. Resultant from start to finish.
Adding and Subtracting Vectors
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Lewin, Walter. 8.01 Physics I: Classical Mechanics, Fall 1999.
(Massachusetts Institute of Technology: MIT OpenCourseWare),
http://ocw.mit.edu (Accessed 22 Feb, 2012). License: Creative
Commons BY-NC-SA
Adding and Subtracting Vectors
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Components: Break down a vector into
component vectors along a set of axes.
y-component of
vector v
vy
cos θ = adj / hyp
= vx / v
vx = v cos θ
v
sin θ = opp / hyp
= vy / v
vy = v sin θ
θ
vx
x-component of
vector v
Adding and Subtracting Vectors

Lewin, Walter. 8.01 Physics I: Classical Mechanics, Fall 1999.
(Massachusetts Institute of Technology: MIT OpenCourseWare),
http://ocw.mit.edu (Accessed 22 Feb, 2012). License: Creative
Commons BY-NC-SA
Adding and Subtracting Vectors
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Add these two vectors using components.
Give the magnitude and direction of the
resultant.
Adding and Subtracting Vectors

Lewin, Walter. 8.01 Physics I: Classical Mechanics, Fall 1999.
(Massachusetts Institute of Technology: MIT OpenCourseWare),
http://ocw.mit.edu (Accessed 22 Feb, 2012). License: Creative
Commons BY-NC-SA
Adding and Subtracting Vectors
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Subtracting = Adding the negative vector
Practice Problem
In which case is the magnitude of vector A+B largest?
In which case is the magnitude of vector A+B smallest?
c
a
Practice Problem
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Two horizontal ropes are attached to a post that is
stuck in the ground. The ropes pull the post
producing the vector forces A = 70 N x̂ + 20 N ŷ and
B = -30 N x̂ + 40 N ŷ as shown in the figure. Find the
direction and magnitude of the horizontal
component of a third force on the post that will
make the vector sum of forces on the post equal to
zero.
Practice Problem
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A Coast Guard ship is located 35 km away
from a checkpoint in a direction 42° north of
west. A sailboat located in still water 20 km
from the same checkpoint in a direction 36°
south of east is about to sink. Draw a
diagram indicating the position of both
ships. In what direction and how far must
the Coast Guard ship travel to reach the
sailboat?
Practice Problem
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Determine the magnitude and direction of
the sum of these four vectors: