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