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Vectors. Chapter 4. Scalar. A quantity with only magnitude. Vector. A quantity with both magnitude and direction. Vector. Tail Head. Resultant Vector. The sum of two or more vectors. Vector Addition. Two addition methods: Graphical Algebraic. Graphical Vector Addition. - PowerPoint PPT Presentation
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VectorsChapter 4
Scalar•A quantity with only magnitude
Vector•A quantity with both magnitude
and direction
VectorTail Head
Resultant Vector•The sum of two or more vectors
Vector Addition•Two addition methods:
•Graphical•Algebraic
Graphical Vector Addition
•Use the following steps
(1)•Draw any one of the vectors with its tail at the starting point or
origin
(2)•Draw the 2nd vector
with its tail at the head of the first
vector
(3)•Draw the resultant
vector from the starting point of the 1st vector to
the head of the 2nd
(4)•Measure the length of
the resultant to determine the
magnitude of the vector
(5)•Measure the angle to determine the direction
of the vector
Drill:• An insect crawls 4.0 cm
east, then 3.0 cm south. Calculate:
• a) distance traveled• b) displacement
Practice:• A plane flies 5.0 km west,
then 2500 m south. Calculate:
• a) distance traveled• b) displacement
Drill:• A bug crawls 3.0 cm west,
then 40.0 mm south. Calculate:
• a) distance traveled• b) displacement
Drill:• A plane flies 150 m/s east
in a 25 m/s wind blowing towards south. Calculate the plane’s velocity relative to the ground.
Review HW•Problems 5 - 10 on page 71
Adding Vectors with Opposite Signs
•Vector1 + (-Vector2) = Vector1 – Vector2
V1V2
V2 - V1
VR
Practice:• A bird flies 25 m west, then
57 m east. Calculate: • a) distance traveled• b) displacement
Practice:• A bird flies 14 m west, then
32 m east, then 21 m west. Calculate:
• a) distance traveled• b) displacement
A boat travels upstream at 10.0 m/s in a river flowing at 2.5 m/s.
Calculate the velocity of the boat.
Multiple vectors•When adding multiple vectors, just repeat the process of head of first to tail of second etc.
Algebraic
A
BR
Practice:• A car goes 3.0 km west,
then 4.0 km south, then 5.0 km north. Calculate:
• a) distance traveled• b) displacement
Algebraic
adj
opphyp
Solving the problem
•Sin = opp/hyp•Cos = adj/hyp•Tan = opp/adj
Algebraic•R2 = A2 + B2 if right angle
•R2 = A2 + B2 –2ABcos otherwise
A ball rolls 45 m north, then is kicked 60.0 m
west. Calculate the distance & displacement
of the ball.
A ball thrown at 50.0 m/s north from a train moving 50.0 m/s west.
Calculate the velocity of the ball.
A boat travels at 4.0 m/s across in a river flowing at 3.0 m/s. Calculate the
velocity of the boat.
A plane travels at 250 m/s south in a 50.0 m/s wind blowing east to west. Calculate the
velocity of the plane.
A plane travels at 25 m/s south in a 15 m/s wind blowing east to west. Calculate the
velocity of the plane.
Drill: A snail travels at 9.0 cm south then 15.0 cm west then 6.0 cm south. Calculate the displacement of the
snail.
Check HW•Problems 11 – 14•Page 74
Vector Resolution•Resolving any vector into its x & y components
Vector = 100 units at 37o N o E
y-axis
x-axis37o
Determine the x & y components
y-axis
Adjacent side37o
Opposite side
Hypotenuse
Solving the problem
•Sin = opp/hyp•Cos = adj/hyp•Tan = opp/adj
Solving the problem
•sin = opp/hyp•opp = hyp x sin
Solving the problem
•cos = adj/hyp•adj = hyp x cos
Determine the x & y componentsy-axis
Adjacent side = hyp(cos )
Opposite side= hyp(sin )
Hypotenuse = 100 m
Trig Functions• x-component = 100(cos 37o)
= 100(0.80) = 80 units
• y-component = 100(sin 37o)= 100(0.60) = 60 units
Resolve the following vector into polar or x
& y components:
150 m/s @ 30o N o E
Resolve the following vector into polar or x
& y components:
250 N @ 37o E o S
Resolve the following vector into polar or x
& y components:
7500 N @ 53o
Vector Addition Hint:• When adding multiple
vectors, just add the vector components. Then solve for the final vector.
1) 50 m at 45o E o N2) 45 m at 53o S o W3) 80 m at 30o W o N4) 75 m at 37o N o ECalculate resultant
Equilibrium•When functions applied to any system add up to zero
•Steady State•Homeostasis
Equilibrant•The vector, when added to a set of vectors, would bring the sum of all the vectors back to the zero point or origin.
An automobile is driven 250 km due west, then 150 km
due south. Calculate the resultant vector.
A dog walks 4.0 miles east, then 6.0 miles
north, then 8.0 miles west. Calculate the
resultant vector.
Drill: A cannon fires a projectile at 37o from horizontal at 1250 m/s
Calculate the x & y components.
Check HW: 11 - 14
A jet flies 15 km due west then 25 km
at 53.1o north of west. Calculate the
resultant vector.
1) 9.0 m W2) 800.0 cm S3) 3000.0 mm E4) 0.0035 km NCalculate equilibrant
Resolve a 2.4 kN force vector that is 30.0o from
horizontal into horizontal & vertical
components in N:
1) 2.0 m at 30o
2) 150.0 cm at 37o
3) 3000.0 mm at 53o
4) 0.0040 km at 127o
Calculate equilibrant
The following forces are acting on a point: 1) 5.0 N at 37o
2) 8.0 N at 53o
Calculate equilibrant
A boat travels at 4.0 m/s directly across a river flowing at 3.0 m/s. Calculate the resultant vector.
A boy walks 4.0 miles east, then 6.0
miles north, then 4.0 miles east. Calculate the resultant vector.
A jet flies 15 km due west then 25 km at 53o north of west.
Calculate the resultant vector.
A jet flies 28 km due west then 21 km north. Calculate the
resultant vector.
A dog walks 8.0 m due east then 15 m at 37o north of east.
Calculate the resultant vector.
A jet travels 250 miles at 37o north of west.
Resolve the displacement into
north & west components.
1) 50 m at 45o E o N2) 45 m at 53o S o W3) 80 m at 30o W o N4) 75 m at 37o N o ECalculate resultant
A girl walks 25 m due east then 15 m at 37o north of east, the 50.0 m due south. Calculate
the resultant vector.
A girl walks 75 m at 37o north of east, then
75 m at 53o west of north. Calculate the
resultant vector.
1) 50 m at 45o S o W2) 75 m at 53o E o S3) 80 m at 37o N o E4) 75 m at 33o W o N
Calculate resultant
Drill: A dog walks:1) 0.16 km due north2) 90.0 m due east3) 25,000 cm at 37o N o E
Calculate: Res. & Eq.
Check HW•Problems 31 & 31
•Page 79
A zombie walks:1) 0.30 km at 30o SoW2) 500 m at 45o NoE
Calculate resultant:
Drill: A snail crawls:1) 25 cm at 37o WoS2) 400 mm at 30o NoE
Calculate resultant:
A telephone pole has a wire pulling with a 3500 N force attached at 20o
from the top of the pole. Calculate the force
straight down.
A cat walks:1) 9.0 m due south2) 1500 cm due east3) 5,000 mm at 37o N o E
Calculate resultant:
Forces act on a point:1) 150 N at 53o EoS2) 250 N at 37o SoW3) 0.50 kN at 45o WoS
Calculate resultant:
1) 350 N at 53o WoS2) 150 N at 37o NoW3) 0.25 kN at 45o WoS4) 250 N due E
Calculate resultant:
1) 0.35 kN due west2) 150 N due south3) 0.50 kN at 45o EoN4) 250 N at 37o NoE
Calculate resultant:
Use graph paper to solve the following:
1) 250 m due east3) 0.50 mm 53o EoN
Calculate resultant:
Drill & Collect HW: Solve the following:
1) 360 m due west3) 0.27 km due north
Calculate resultant:
HW: Solve with trig:1) 0.10 N 37o SoW2) 250 kN 53o EoN3) 150,000 N East
Calculate resultant:
Use graph paper to solve the following:
1) 3.0 m due west3) 15 m 53o EoN
Calculate resultant:
1) 0.35 km due west2) 250 m due south3) 0.50 km at 45o EoN4) 150 m at 37o NoE
Calculate resultant:
Define the Following:•Scalar•Vector•Magnitude•Direction
Define the Following:•Distance•Displacement•Speed•Velocity
Test Review
Terms to Define:• Equilibrant• Vector Resultant• Scalar• Vector• Vector Resolution
Metric Prefixes:•Centi Kilo•Giga Mega•Micro Milli•Nano
Trig Functions:•Sin Pytha-•Cos Theorem•Tan •Law of Cosines
Add the 3 Vectors Graphically:
•50.0 m west•90.0 m north•170 m east
Add the 2 Vectors Mathematically:
•20.0 m west•0.10 km @ 37oNoE
Resolve the Vector into x & y comp:
•0.450 km @ 53o SoW
Add the 3 Vectors using vector components:
•75 m @ 37o NoW•90.0 m @ 37o NoE•150 m @ 53o SoW