Post on 02-Jan-2016
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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
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
A
A
A
A
A
Vectors have magnitude and direction,but no place
Vectors ad Their Properties
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
A
B
Two vectors are equal if they have the same magnitude
and the same direction
BA
Vectors ad Their Properties
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
Scaling Vectors
A
A2
A23
Vectors ad Their Properties s
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
Scaling a Displacement Vector
d
td v
Vectors ad Their Properties
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
Scaling a Change in Velocity Vector
v
tv a
Vectors ad Their Properties
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
Vector Addition (Graphical)
A
B
BAC
Vectors ad Their Properties
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
Vector Subtraction (Graphical) (A B)
A BB
Vectors ad Their Properties
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
C
BAC
BAC
A
x
y A A
cosAAx
sinAAy
Vector Components
Components of a Vector
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
x
A A
A
y
yA
xA
2y
2x
2 AAA
2y
2x AAA
x
y1
A
Atan
Vector Components
Components of a Vector
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
A
BR
AxBx
Rx
Ay
By
Ry
x
y
2y
2x RRR
xxx BAR
yyy BAR
Components of a Vector
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
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
Components of a Vector
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
F2
2
F3
3 F4
4
n
1kkR FF
4321R FFFFF
F1
1
Components of a Vector
Adding Vectors
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
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
Components of a Vector
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
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
Components of a Vector
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
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
Components of a Vector
Add the following vectors
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
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.
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
Motion in Two Dimensions
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
Motion in Two Dimensions
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
Ignoring air resistance, the horizontal component of a projectile's velocity
A) is zero.
B) remains constant.
C) continuously increases.
D) continuously decreases.
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
Motion in Two Dimensions
vo
x
y At t = 0
atvv o
t gvv yoy
gtθsinvv oy Eq 2
Constant acceleration
θsinvv oyo
Eq 1 θcosovxv
Constant velocity
Motion in Two Dimensions
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
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
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
Motion in Two Dimensions
vo
x
y
SubEq 1
t cosvx o Eq 3
1 Eq. cosvv ox
tΔxΔ
v
tx
vx
tx
θcosvo
Constant velocity
Motion in Two Dimensions
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
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
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
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
Motion in Two Dimensions
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
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
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
Motion in Two Dimensions
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?
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
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
Motion in Two Dimensions
Vertical Position as a Function of Horizontal Displacement
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
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
Motion in Two Dimensions
Maximum Height
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
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.
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
Motion in Two Dimensions
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?
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
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
Motion in Two Dimensions
Range
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
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°
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
Motion in Two Dimensions
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).
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
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
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
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
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
vw
d
vw
vS2B
2wS vvv vB
B
wv
v
Ld
Bvwv
Ld
vBL
Relative Velocity
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
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
Vectors and Two-Dimensional Motion (Phy 2053): vittitoe
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 (Phy 2053): vittitoe
Motion in Two Dimensions