Vector Addition

Recall that for vectors in one dimension (parallel vectors), the vectors are added algebraically.

Vectors in 2 dimensions are add geometrically

RY

θ

X

*X = (R) (cos θ)

*Y = (R) (sin θ)

Example

3A:Section Review Pg. 87 1, 2 2.) 126m 10o= θ

RY

θ

Vector Resultants with Right Triangles

c2 = a2 + b2 [R2 = x2 + y2 ]

θ

R

X

7 km

h

6km/h

R2 = (6km/h) 2 + (7km/h)2

R2 = 36km2/h2 + 49km2/h2

R = 85 km2/h2

R = ___________ km/h

θ– theta: means the measure of the angle

We will not do this vector addition graphically. We will not do it using the Law of Cosines

→We will do it trigonometrically or as the books says, analytically.

All the Trig. You need to know for Physics I

*SOH, CAH, TOA

hypotenuse

Side adjacent to θ

θ

Side opposite of θ

Sin θ = Opp. Side Hypotenuse

Cos θ = Adjacent side Hypotenuse

Tan θ = Opposite side Adjacent side

Solving Problems with 2 vectors in different directions:

ExampleSample 3B page 93*Add to your notes

Vr2 = Vx

2 + Vy2

Angles are in Standard PositionWhat you learned in math is used in Physics!

0o

270o

900

1800

θ

50km/h

90 km/h

Example:

?

Finding the resultant without R 2 = X 2 + Y 2

R = Resultant

Tan θ = 50 km/hr = .5556 90 km/hr

=θ 29.05o

Cos 29.05o = 90km/hrR

R = 90 km/hr .8742

R = 102.9 km/hr

R = 102.9 km/hr @ 29.05o STD position

I

II

Motion in 2 dimensionsI. Projectiles: Objects thrown or projected into the air are projectiles

whose parabolic paths are called trajectories.

Motion of a projectile along a trajectory is called projectile motion.

Max Height

*Vx = (Vi) (cos θ)

*Vy = (Vi) (sin θ)

Once a projectile loses contact with the hand, bat, or gun barrel,

it is only accelerated by g, in the X direction, a = 0

θVi

Vx

Vy

Motion in 2 dimensions

Max Height

*Vx = (Vi) (cos θ)

*Vy = (Vi) (sin θ)

We can analyze the motion separately along each axis:

X direction Y directionX = (Vx)(t) + (½)at2 = (Vx)(t) X = (Vx)(t) = (Vi cos θ)t

θY

X

Motion in 2 dimensions

Max Height

*Vx = (Vi) (cos θ)

*Vy = (Vi) (sin θ)

We can analyze the motion separately along each axis:

X direction Y directionX = (Vx)(t) + (½)at2 = (Vx)(t)

θY

X

Motion in 2 dimensions

Max Height

*Vx = (Vi) (cos θ)

*Vy = (Vi) (sin θ)

We can analyze the motion separately along each axis:

X direction Y directionX = (Vx)(t) + (½)at2 = (Vx)(t) Y = (Vy)(t) + ½gt2

X = (Vx)(t) = (Vi cos θ)t

θY

X

Motion in 2 dimensions

Max Height

*Vx = (Vi) (cos θ)

*Vy = (Vi) (sin θ)

We can analyze the motion separately along each axis:

X direction Y directionX = (Vx)(t) + (½)at2 = (Vx)(t) Y = (Vy)(t) + ½gt2

X = (Vx)(t) = (Vi cos θ)t Y = (Vy)(t) + ½gt2 = (Vi sin θ)t + ½gt2

* t = ½ time of flight for Y

θY

X

*Time of Flight Equation: (derived from the Y equation)

Y = vi sinθ (t) + ½ gt2

2vi sin θ = tg

or

2vy = t g

Calculating Xmax without time:

X max = 2 Vi 2 sinθ cos θ

g

X max = 2 Vi 2 sin 2 θ

g

Example : A stone is thrown horizontally at 15m/s from the top of a cliff that is 44 m high.

A. How long does the stone take to reach the bottom of the cliff? B. How far from the base of the cliff does the stone strike the ground?C. Sketch the trajectory of the stone.

Acceleration only vertically means:the time aloft is independent of the horizontal

component of the trajectory.

Max height = max Y

Same t: time

*if vi is the same

Fly outPop upY

Homework #1 BK & WKBK A-C Chp. 3 (15)

Book Sections 3A,3B & 3C

3A: Pg. 91 1-42.) 45.6m at 9.5o East of North4.) 1.8m at 49o below the horizontal

3B: pg. 94 1,3,5 (answers back of the book)

3C: pg. 97 1,2,42.) 7.5km at 26o above the horizontal4.) 171 km at 34o East of North

WKBK 3A & 3B3A:

3.) 7 jumps at 36o West of North 5.) 65o below the water 533m

3B 4.) x= 29m y=21m5.) Vx=335km/h Vy=89.8km/h6.) d= 900m

x=450 m East y= 780m North

Workbook 3 D-E

3D 1. 5.98 m = X3. 170m = Y4. y= -.735m

3E 1. Vi = 45.8 m/s2. V=Vx= 68.2 m/s4. Vi=20.8 m/s

Ymax = 11m Ymax = 22.1m

Homework #2 Chp. 3 Book & WKBK 3D-3E (11)

BOOK-3D Problems pg. 102 2,3,42.) 4.9m/s3.)7.6m/s4.) 5.6m

3E problems pg. 104 3 & 4 4.) 7m/s

*Scalars – has magnitude with units

*Vectors – have magnitude with direction

→ Vectors can be added and subtracted creating a Resultant ( R )

3N 4N

7N

6N

3N

3N