vectors and two-dimensional motion. 3-01 vectors and their properties 3-02 components of a vector...

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x

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Vectors andTwo-Dimensional Motion

3-01 Vectors and Their Properties

3-02 Components of a Vector

3-04 Motion in Two Dimensions

Vectors & Two-Dimensional Motion

3-05 Relative Velocity

Topics

Vectors and Two-Dimensional Motion (Phy 2053): vittitoe

Scalars Vectors

Distance

Time

Mass

Work

Energy

Speed

Displacement

Velocity

Acceleration

Force

Momentum

Torque

MagnitudeOnly

Magnitudeand

Direction

Vectors ad Their Properties


A

A

A

A

A

Vectors have magnitude and direction,but no place



A

B

Two vectors are equal if they have the same magnitude

and the same direction

BA



Scaling Vectors

A

A2

A23

Vectors ad Their Properties s


Scaling a Displacement Vector

d

td v



Scaling a Change in Velocity Vector

v

tv a



Vector Addition (Graphical)

A

B

BAC



Vector Subtraction (Graphical) (A B)

A BB



C

BAC

BAC

A

x

y

xA

yA yx AAA

Components of a Vector


A

x

y A A

cosAAx

sinAAy

Vector Components



x

A A

A

y

yA

xA

2y

2x

2 AAA

2y

2x AAA

x

y1

A

Atan

Vector Components



A

BR

AxBx

Rx

Ay

By

Ry

x

y

2y

2x RRR

xxx BAR

yyy BAR



2y

2x RRR

xxx BAR

yyy BAR

A

BR

Ax Bx

Rx

Ay

By

Ry

x

y

α

β

βcosBB

αcosAA

x

x

sinBB

sinAA

y

y



F2

2

F3

3 F4

4

n

1kkR FF

4321R FFFFF

F1

1


Adding Vectors


F1F2

F3

F4

1

2

3 4

F1 = 50 N 1 = 30o

F2 = 100 N 2 = 135o

F3 = 30 N 3 = 250o

F4 = 40 N 4 = 300o

θsinF θcosF kkkk

43.3 25.070.7 70.710.3 28.220.0 34.6

17.7 32.9

22R 9.327.17F

N 4.37FR



22R 9.327.17F

N 4.37FR

7.17

9.32tan

7.17

9.32tan 1 o7.61

FR = 37.4 N

R

17.7

32.9

180R 7.61180

oR 118



F1F2

F3

F4

1

2

3 4

F1 = 90 N 1 = 45o

F2 = 80 N 2 = 150o

F3 =50 N 3 = 220o

F4 = 70 N 4 = 340o


Add the following vectors


Motion in Two Dimensions


ay = g

ax = 0



Ignoring air resistance, the horizontal component of a projectile's acceleration

A) is zero.

B) remains a non-zero constant.

C) continuously increases.

D) continuously decreases.



If an object is launched at an initial angle of θ with the horizontal, the analysis is similar except that the initial velocity has a vertical component.

xv

yv

ovxv

yv1v

xv2v

xv

yv3v

xv

yv

fv

31 vv

fo vv



Ignoring air resistance, the horizontal component of a projectile's velocity

A) is zero.

B) remains constant.

C) continuously increases.

D) continuously decreases.



vo

x

y At t = 0

atvv o

t gvv yoy

gtθsinvv oy Eq 2

Constant acceleration

θsinvv oyo

Eq 1 θcosovxv

Constant velocity



A ball is thrown with a velocity of 20 m/s at an angleof 60° above the horizontal. What is the horizontal component of its instantaneous velocity at the exact top of its trajectory?

A) 10 m/s

B) 20 m/s

C) 5.0 m/s

D) zero



vo

x

y

SubEq 1

t cosvx o Eq 3

1 Eq. cosvv ox

tΔxΔ

v

tx

vx

tx

θcosvo

Constant velocity



Horizontal Position

EOC Problem 03-04

If Vx = 6.80 units and Vy = 7.40 units, a) determine the magnitude of V.

b) determine the direction of V

Vx

VVy


vo

x

y

2

gtt θsinvy

2

o Eq 4

2at

tvxx2

oo

Constant acceleration

2

t gtv0y

2

yo

2

t gt sinvy

2

o

θsinvv oyo



Vertical Position

A soccer ball is kicked with a velocity of 25 m/s at an angle of 45° above the horizontal. What is the vertical component of its acceleration as it travels along its trajectory?

A) 9.80 m/s2 downward

B) (9.80 m/s2) × sin (45°) downward

C) (9.80 m/s2) × sin (45°) upward

D) (9.80 m/s2) upward



A diver running 1.8 m/s dives out horizontally from the edge of a vertical cliff and 3.0 s later reaches the water below. a) How high was the cliff?

EOC Problem 03-18

h

d

v

b) How far from its base did the diver hit the water?


voh

x

y

Sub into Eq 4 θcosv

xt

o

2

ooo θ cosv

xg2

1θsinθ cosv

xvy

θcosv2

gxθtan xy

22o

2 Eq 5

3 Eq. t θcosvx o

4 Eq. 2

gtt θsinvy

2

o

Solve Eq 3 for t


Vertical Position as a Function of Horizontal Displacement


voh

x

y

At the maximum

height (vy = 0)

0gtsinvv oy

Sub into Eq 4

g2

sinv y

22o

max

Eq 7

4 Eq. 2

gtt sinvy

2

o

2 Eq. gtsinvv oy

g

sinvt otop

Eq 6

2i

io

gsinv

2g

gsinv

sinvy


Maximum Height


When a football in a field goal attempt reaches its maximum height, how does its speed compare to its initial speed?

A) It is zero.

B) It is equal to its initial speed.

C) It is greater than its initial speed.

D) It is less than its initial speed.



EOC Problem 03-22

A football is kicked at ground level with a speed of 18.0 m/s at an angle of 35.0º to the horizontal. How much later does it hit the ground?


vo

R

hx

y

Eq.6 g

θsinvt otop

g

θsinv2t2t otopair

airxtvR

g

θsinv2θcosvR o

o

g

θ2sinvR

2o Eq 8

1 Eq. θcosvv ox


Range


At what angle should a water-gun be aimed in order for the water to land with the greatest horizontal range?

A) 0°

B) 30°

C) 45°

D) 60°



A projectile is fired with an initial speed of 65.2 m/s at an angle of 34.5º above the horizontal on a long flat firing range. Determine (a) the maximum height reached by the projectile.

EOC Problem 03-30

(b) the total time in the air

(c) the total horizontal distance covered (that is, the range).


cosvv ox Eq 1Horizontal Velocity

gtsinvv oy Eq 2Vertical Velocity

t cosvx o Eq 3Horizontal Displacement

2

gtt sinvy

2

o Eq 4Vertical Displacement

Equations


22

o

2

cosv2

gxtan xy Eq 5

Vertical Position

g2

sinv hy

22o

Eq 7Maximum Height

g

2sinvR

2o

Eq 8Range

g

sinvt otop

Eq 6Time to

the Top

Equations


vw

d

vw

vS2B

2wS vvv vB

B

wv

v

Ld

Bvwv

Ld

vBL

Relative Velocity


vS

L

2S

2W

2B vvv

2W

2BS vvv

B

W1

v

vsin

tL

vs

svL

t

vB

vw

Time required to go directly across.

2W

2B vv

L

Relative Velocity


You are trying to cross a river that flows due south with a strong current. You start out in your motorboat on the east bank desiring to reach the west bank directly west from your starting point. You should head your motorboat

A) due west.

B) due north.

C) in a southwesterly direction.

D) in a northwesterly direction.



vectors and two-dimensional motion. 3-01 vectors and their properties 3-02 components of a vector...

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