Vectors You will be tested on your ability to: 1.correctly express a vector as a magnitude and a...

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Vectors You will be tested on your ability to: 1. correctly express a vector as a magnitude and a direction 2. break vectors into their components 3. add and multiply vectors 4. apply concepts of vectors to linear motion equations (ch. 2)

Transcript of Vectors You will be tested on your ability to: 1.correctly express a vector as a magnitude and a...

Page 1: Vectors You will be tested on your ability to: 1.correctly express a vector as a magnitude and a direction 2. break vectors into their components 3.add.

Vectors

You will be tested on your ability to:1. correctly express a vector as a

magnitude and a direction2. break vectors into their components

3. add and multiply vectors4. apply concepts of vectors to linear

motion equations (ch. 2)

Page 2: Vectors You will be tested on your ability to: 1.correctly express a vector as a magnitude and a direction 2. break vectors into their components 3.add.

Vector vs. Scalar

• Scalar units are any measurement that can be expressed as only a magnitude (number and units)– Examples:

• 14 girls• $85• 65 mph

• Vector quantities are measurements that have BOTH a magnitude and direction.– Examples:

• Position• Displacement• Velocity• Acceleration• Force

Page 3: Vectors You will be tested on your ability to: 1.correctly express a vector as a magnitude and a direction 2. break vectors into their components 3.add.

Representing Vectors

• Graphically, vectors are represented by arrows, which have both a length (their magnitude) and a direction they point.

v = 45 m/s

v = 25 m/s d = 50 m

a= 9.8 m/s2

Page 4: Vectors You will be tested on your ability to: 1.correctly express a vector as a magnitude and a direction 2. break vectors into their components 3.add.

Representing Vectors

• The symbol for a vector is a bold letter.– For velocity vectors we write v– For handwritten work we use the letter with an

arrow above it. v

• Algebraically– Vectors are written as a magnitude and

direction– v = lvl , Θ– Example v = 25 m/s, 120o or d = 50 m, 90o

Page 5: Vectors You will be tested on your ability to: 1.correctly express a vector as a magnitude and a direction 2. break vectors into their components 3.add.

Drawing Vectors

• Choose a scale• Measure the direction of the vector starting with east as

0 degrees. • Draw an arrow to scale to represent the vector in the

given direction• Try it! v = 25 m/s, 190o scale: 1 cm = 5 m/s• This can be described 2 other ways

– v = 25 m/s, 10o south of west– v = 25 m/s, 80o west of south

• Try d = 50 m, 290o new scale?

Page 6: Vectors You will be tested on your ability to: 1.correctly express a vector as a magnitude and a direction 2. break vectors into their components 3.add.

Adding Vectors

• Vector Equation– vr = v1+v2

– Resultant- the vector sum of two or more vector quantities.

– Numbers cannot be added if the vectors are not along the same line because of direction!

– Example……– To add vector quantities that are not along the

same line, you must use a different method…

Page 7: Vectors You will be tested on your ability to: 1.correctly express a vector as a magnitude and a direction 2. break vectors into their components 3.add.

An Example

D1

D2

D3

DT

D1 = 169 km @ 90 degrees (North)

D2 = 171 km @ 40 degrees North of East

D3 = 195 km @ 0 degrees (East)

DT = ???

Page 8: Vectors You will be tested on your ability to: 1.correctly express a vector as a magnitude and a direction 2. break vectors into their components 3.add.

Tip to tail graphical vector addition

• On a diagram draw one of the vectors to scale and label it.

• Next draw the second vector to scale, starting at the tip of the last vector as your new origin.

• Repeat for any additional vectors• The arrow drawn from the tail of the first vector

to the tip of the last represents the resultant vector

• Measure the resultant

Page 9: Vectors You will be tested on your ability to: 1.correctly express a vector as a magnitude and a direction 2. break vectors into their components 3.add.

Add the following

• d1 = 30m, 60o East of North

• d2 = 20m, 190o

• dr = d1+d2

• dr = ?

• dr = 13.1m, 61o

Page 10: Vectors You will be tested on your ability to: 1.correctly express a vector as a magnitude and a direction 2. break vectors into their components 3.add.

Vector Subtraction• Given a vector v, we define –v to be the same

magnitude but in the opposite direction (180 degree difference)

• We can now define vector subtraction as a special case of vector addition.

• v2 – v1 = v2 + (-v1) • Try this• d1 = 25m/s, 40o West of North• d2 = 15m/s, 10o

• 1cm = 5m/s• Find : dr = d1+d2

• Find dr = d1- d2

v–v

Page 11: Vectors You will be tested on your ability to: 1.correctly express a vector as a magnitude and a direction 2. break vectors into their components 3.add.

• Multiplying a vector v by a scalar quantity c gives you a vector that is c times greater in the same direction or in the opposite direction if the scalar is negative.

V

cV

-cV

Page 12: Vectors You will be tested on your ability to: 1.correctly express a vector as a magnitude and a direction 2. break vectors into their components 3.add.

Vector Components

• A vector quantity is represented by an arrow.• v = 25 m/s, 60o

• This single vector can also be represented by the sum of two other vectors called the components of the original.

v =

50 m

/s, 6

0o

sinΘ = Vy / V

Vy= V sinΘ

cosΘ = Vx / V

Vx= V cosΘ

Page 13: Vectors You will be tested on your ability to: 1.correctly express a vector as a magnitude and a direction 2. break vectors into their components 3.add.

Try this:V1 = 10 m @ 30 degrees above +x

Find: V1X = V1Y =

V2 = 10 m @ 30 degrees above –x

Find: V2X = V2Y =

But V2X should be

NEGATIVE!!!

Try using the angle 150 degrees for V2

Ө1Ө2

Page 14: Vectors You will be tested on your ability to: 1.correctly express a vector as a magnitude and a direction 2. break vectors into their components 3.add.

Try this:V1 = 10 m @ 30 degrees above +x

Find: V1X = V1Y =

V2 = 10 m @ 30 degrees above –x

Find: V2X = V2Y =

Find : V3X =V3Y =

V3 = 10 m @ 30 degrees below +x

Try using the angle 330 degrees for V3

Page 15: Vectors You will be tested on your ability to: 1.correctly express a vector as a magnitude and a direction 2. break vectors into their components 3.add.

Now try this:

VX = 25m/sVY = - 51m/s

Find V=

Page 16: Vectors You will be tested on your ability to: 1.correctly express a vector as a magnitude and a direction 2. break vectors into their components 3.add.

and your point is???

• ALWAYS:

• ALWAYS:

• ALWAYS:

Describe a vector’s direction relative to the +x axis

Measure counter-clockwise angles as positive

Measure clockwise angles as negative

Page 17: Vectors You will be tested on your ability to: 1.correctly express a vector as a magnitude and a direction 2. break vectors into their components 3.add.

An Example

D1

D2

D3

DT

D1 = 169 km @ 90 degrees (North)

D2 = 171 km @ 40 degrees North of East

D3 = 195 km @ 0 degrees (East)

DT = ???

D2X

D2Y

D1Y

D3X

Page 18: Vectors You will be tested on your ability to: 1.correctly express a vector as a magnitude and a direction 2. break vectors into their components 3.add.

A Review of an Example

y (km)

x (km)

Page 19: Vectors You will be tested on your ability to: 1.correctly express a vector as a magnitude and a direction 2. break vectors into their components 3.add.

But Wait. . . There’s more!

y (km)

x (km)

We’ve Found:

DTX = 326 km

DTY = 279 km.

For IDTI, use the Pythagorean Theorem.

For the Direction of DT, use Tan-1

Page 20: Vectors You will be tested on your ability to: 1.correctly express a vector as a magnitude and a direction 2. break vectors into their components 3.add.

Practice it:

• Pg. 70, # 1, 4