VECTORSVECTORS Vectors and Scalars A vector has magnitude as well as direction. Some vector quantities: displacement, velocity, force, momentum.

VECTORS

Vectors and ScalarsA vector has magnitude as well as direction.Some vector quantities: displacement, velocity, force, momentumA scalar has only a magnitude.Some scalar quantities: mass, time, temperature

Addition of Vectors—Graphical MethodsFor vectors in one dimension, simple addition and subtraction are all that is needed. You do need to be careful about the signs, as the figure indicates.

Vector -

Vector Addition Graphically – Finding ResultantResultant – Diagram

Diagram

“Vectors can be __________________ as long as

‘_____________________’ and ‘___________________’ are unchanged”

Concurrent Diagram Head To Tail Diagram

NO!!!

Find Resultant: 1) Arrange vectors head to tail (ORDER DOES NOT MATTER)

2) Draw resultant toward end of last vector (DON’T FORGET ARROW!)

3) Measure length of resultant and apply scale to find magnitude.

4) Use protractor to determine direction

AKA: Parallelogram Method

Scale: 1cm = 5m/s

Standard System: 0 - 360°Final Answer: IF USING POLAR COORDINATES…

Even if the vectors are not at right angles, they can be added graphically by using the “tail-to-tip” method.

Use of Pythagorean Theorem and TrigonometryMagnitude

Direction SOH CAH TOAR

35m

25m

θ

Vector Addition Analytically – Finding Resultant

Resultants vs. Equilibrants

Resultants vs. Equilibrants

As θ between two vectors goes from 180º to 0º…resultant ___________equilibrant ___________

Problem: Determine the distance between two people both graphically and analytically…

A puppy is stranded on a rock in a river flowing East at 3m/s. In an attempt to rescue the puppy, you swim North at 1m/s. If the puppy is 10m across the river...

 a) What will be your resultant velocity?

 b) How far upstream would you have to start swimming in order to reach the puppy?

1. For one hour, you travel east in your car covering 100 km .Then travel south 100 km in 2 hours. You would tell your friends that your average speed wasA. 47 km/hrB. 67 km/hrC. 75 km/hrD. 141 km/hrE. 200 km/hr

2. For one hour, you travel east in your car covering 100 km .Then travel south 100 km in 2 hours. You would tell your friends that your average velocity was

A. 47 km/hrB. 67 km/hrC. 75 km/hrD. 141 km/hrE. 200 km/hr

3. You have already traveled east in your car 100 km in 1 hr and then south 100 km in 2 hrs. To get back home, you then drive west 100 km for 3 hours and then go north 100 km in 4 hours. You would say your average velocity for the total trip was

A.20 km/hrB.40 km/hrC.60 km/hrD.100 km/hrE.None of the above

Vector ComponentsAny vector can be expressed as the sum of two other vectors, which are called its components. Usually the other vectors are chosen so they are perpendicular to each other.

Vector Components Graphically

V = 10

m/s

Y- Component: Vy - drop vertical line to X – axisX- Component: Vx - fill in horizontal line* Arrows should lead to end of resultantScale:Vx =Vy =

Vector Components AnalyticallyA flare is shot at 300m/s at 60°. How fast was it traveling horizontally and vertically at the moment it was shot.

Vx =

Vy =

60°

300 m/s

cosθ = Vx

V

sinθ = Vy

V

Adding Vectors by ComponentsThe components are effectively one-dimensional, so they can be added arithmetically:

Adding Vectors by ComponentsAdding vectors:1. Draw a diagram; add the vectors graphically.2. Choose x and y axes.3. Resolve each vector into x and y components.4. Calculate each component using sines and cosines.5. Add the components in each direction.6. To find the length and direction of the vector, use:© 2014 Pearson Education, Inc.

Example: Adding Vectors by ComponentsDetermine the magnitude and direction of the resultant for the following 3 vectors…Sketch a vector diagram showing the 3 vectors, and the components of each…A = 8.5mi @ 45°B = 8.5mi @ 135°C = 30mi @ 0°Vector X – comp Y – comp

A

B

C

Resultant

Example: Firing a Slingshot

1. You fly east in an airplane for 100 km. You then turn left 60 degrees and travel 200 km. How far east of the starting point are you? (approximately) A.100 kmB.150 kmC.200 kmD.300 kmE.none of the above

2. You fly east in an airplane for 100 km. You then turn left 60 degrees and fly 200 km. How far north of the starting point are you? (approximately) A.100 kmB.130 kmC.170 kmD.200 kmE.none of the above

3. You fly east in an airplane for 100 km. You then turn left 60 degrees and fly 200 km. How far from the starting point are you? (approximately) A.170 kmB.200 kmC.260 kmD.300 kmE.370 km

4. You fly east in an airplane for 100 km. You then turn left 60 degrees and fly 200 km. In what direction are you from the starting point?A. South of westB. Directly southwestC. Directly northeastD. North of eastE. None of the above