Vectors

A B

Vector Definition

Any measurement which includes both size and direction

10 m/s isn’t a vector

25 m/s [SW] is a vector

Size and Scale

The resultant of a vector addition can be measured or calculated using scale drawings or Pythagorean Theorem

Vector Addition [0º 0r 180º]

Two vectors pointing in the same direction are simply added [direction same]

Two vectors pointing in opposite directions are simply subtracted [direction of larger vector]

+

+

5 m [E] 3 m [E]

= 8 m [E]

5 m [E] 3 m [W]

= 2 m [E]

Vector Addition [900]

To add vectors at 90º use a drawing or Pythagoras Theorem

c = (a2 +b2) = (3.02 +4.02) = 5.0 cm = tan-1(E or W vector/N

or S vector) = tan-1(3.0 / 4.0) =37o

r = 5.0 cm

[37o ]

a =

4.0

cm

[N

] b = 3.0 cm [E]

Directions

Use the compass rose to the left to calculate the direction of a vector.

Find angle and then transform it according to quadrant.

N,0º

E,90 º

S,180 º

W,270 º

NE quadrant:just find angle

SE quadrant:180 - angle

SW quadrant:180 + angle

NW quadrant:360 - angle

Example Problem

5 N [S] + 12 N [W] = 13 N [SW]

Resolving Vectors

A vector may be “resolved” into 2 right –angled ( orthogonal) components. This technique can be used to add vectors at odd angles together.

Example: Resolve 15 m/s [235o] into components along compass axes.

1. Determine the quadrant• SW

2. Calculate acute angle• 55o

3. Calculate magnitude of components• 15 sin55o = 12 m/s [W]• 15 cos55o = 9 m/s [S]

15 m/s [235o ]

9 m

/s [

S]

12 m/s [W]

=55

0

Example 2: Add 15 m/s [235o] + 35 m/s [3560] by resolving into components along compass axes

and then adding components.

15 m/s [235o] Acute angle = 55o

15cos55 = 9 ms-1 [S] 15sin55 = 12ms-1 [W] 35 m/s [3560] Acute angle = 4o

35cos4 = 35 ms-1[N] 35sin4 = 2 ms-1 [W] Resultant =12 ms-1 [W] + 9 ms-1 [S]

+ 35 ms-1 [N] + 2 ms-1 [W]• = 14 m/s [W] + 26 m/s [N]• = 30 m/s [NW]

55o

4o

Classwork (8 bonus pts):

25.5 N [129o] +36.7 N [322o] =• 25.5 N [129o] = 19.8 N [E] + 16.0 N [S]• 36.7 N [322o] = 22.6 N [W] + 28.9 N [N]• 28.9 N [N] + 16.0 N [S] = 12.9 N [N]• 19.8 N [E] + 22.6 N [W] = 2.8 N [W]

• 12.9 N [N] + 2.8 N [W] = 13.2 N [N]

Lab # 7

Your mission is to “fly” around the country (minimum 10 trips) to find a possible site for a SCICORP regional office.

Come home when you’re done!

Click on map to retrieve assignment

Scale

On the map, the scale indicates that 2.02 cm = 500 km

This means that 1 cm = _____ km

250 km

Lab 7

Log your flights using the lab 7 word document in the physics assignments folder

Compile the itinerary underneath the map

DEP city ARR city N or S d (cm) E or W d (cm) Resultant (cm) Resultant (km) DirectionWashington, DC Little Rock, AK

Total d (km)

First Trip: DC to Little Rock

• Draw vector from DC to Little Rock. Find size by right clicking on it and choosing “format auto-shape”

• 2.55 cm [S] + 5.13 cm [W]• = (2.552 +5.132)• = 5.73 cm• Next, convert to km using scale

1 cm = 250 km• = 5.73*250 = 1 430 km• Calculate direction (use notes)• SW

•Now make 10 sequential trips around the country. (Bonus points for >20 trips)