Chapter 3 2D Motion and Vectors
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Chapter 3
2D Motion and Vectors
2D Motion and Vectors
• Introduction to Vectors• Vector Operations• Projectile Motion• Relative Motion
Scalar vs. VectorScalar Vector
Definition A quantity that has magnitude but no direction
A quantity that has both magnitude and direction
Examples
Notation v v
-Distance-Speed-
-Displacement-Velocity-
• Walk 7 squares east• Walk 5 squares south• Walk 3 squares west• Walk 2 squares north
Start
End
7
5
3
2
7
5
3
2
While driving through the city, you drive 3 blocks south, 5 blocks east, 5 blocks north, 7 blocks east, 4 blocks south, 2 blocks east, and 2 blocks north.
What is your total displacement?What is the total distance traveled? 14 blocks
28 blocks
�⃗� �⃗� 𝐶 �⃗��⃗�
�⃗� �⃗��⃗�
𝐼
Which vectors have the same direction?
Which vectors have the same magnitude?
Which vectors are identical?
A, H B, F D, E C, I
A, B, D, H C, G, I E, F
A, H C, I
Multiplying and Dividing by a Scalar
�⃗�
2 �⃗�
�⃗� �⃗�/2
= ?
= ?
Adding Vectors
�⃗��⃗�
�⃗�
�⃗�
�⃗�
�⃗�
�⃗�+�⃗�
�⃗�+ �⃗�
�⃗�+�⃗�
�⃗�+ �⃗�
Subtracting Vectors – “Add negative”
= ?
= ?
�⃗�
�⃗� �⃗�
− �⃗��⃗�− �⃗�
− �⃗�
�⃗�
�⃗�− �⃗�
)
)
Subtracting Vectors – “Fork”
= ?
= ?
�⃗�
�⃗��⃗�− �⃗�
�⃗�− �⃗�
�⃗�
�⃗�
�⃗�
�⃗�
�⃗�− �⃗�
�⃗�− �⃗�
Trig Review
a=4
b=3
Pythagorean Theorem
Angles𝑐 2=𝑎 2+𝑏2
c
θ
x
yResultant Vectors
A=7 cm
B=5 cm
Magnitude
Direction
= 8.6 cmθ
= 35.5° NE
R
While following a treasure map, a pirate walks 7.50 m east and then turns and walks 45.0 m south. What single straight-line displacement could the pirate have taken to reach the treasure?
xy
θ
45.0 m
7.50 m
R
= 45.6 m
= 80.5° SE
Components of Vectors
A Ay
Ax = ?Ay = ?
x
y
Ax
A
Practice: A = 5.0 cm, θ = 53.1°
θ x
y
Ay
Ax
“Squished” → sin“Collapsed” → cos
↑Not always the case!
B
= 6.0 m, 60° WN Bx = ?By = ?
= –5.2 mx
y
= 3.0 m
θ By
Bx
x
y
θ
1500 km
A plane flew 25.0° west of south for 1500 km. How far would it have traveled if it flew due west and then due south to get to its destination?
Distance traveled west: =
=
Distance traveled south: =
= Total distance: =
Adding Non-perpendicular Vectors
• Break down each vector into its x- and y-components
• Add all of the x-components • Add all of the y-components• Calculate the resultant vector
Bx
By
A
B
= ?
x
5
7
12
y
0
5
5
A
B
A+B
R
θ
Bx
By
Ax
AyA B
x
5
7
12
y
–3
5
2
A
B
A+B
= ?
Rθ
x
y
Ax
Ay
= 15.0 m, 40° NE = 7.0 m, 15° WN = ?
A
B
θA
x
= 11.5
= 1.81
9.69 m
y
= 9.64
= 6.76
16.4 mR
A
B
A+B
θ
θB
Bx
By
A pilot’s planned course is to fly at 150 km/hr at 30° SW. If the pilot meets a 25 km/hr wind due east, how fast does the plane travel, and in what direction?
x
y
θ
plane
wind
x
=
y
=
x
y
plane
wind
total
x
y
totalθ
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