2-d motion
DESCRIPTION
2-D motion. Scalars and Vectors. A scalar is a single number that represents a magnitude Ex. distance, mass, speed, temperature, etc. A vector is a set of numbers that describe both a magnitude and direction Ex. velocity (the magnitude of velocity is speed), force, momentum, etc. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/1.jpg)
2-D motion
![Page 2: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/2.jpg)
2
Scalars and Vectors
• A scalar is a single number that represents a magnitude– Ex. distance, mass, speed,
temperature, etc.
• A vector is a set of numbers that describe both a magnitude and direction– Ex. velocity (the magnitude of velocity
is speed), force, momentum, etc.
![Page 3: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/3.jpg)
3
Scalars and Vectors
•Notation: – a vector-valued variable will be
Bold, – a scalar-valued variable will be in
italics. – if hand written vectors can be
denoted by an arrow over the value.
a
![Page 4: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/4.jpg)
4
Characteristics of Vectors
A Vector is something that has two and only two defining characteristics:
1. Magnitude: the 'size' or 'quantity'
2. Direction: the vector is directed from one place to another.
![Page 5: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/5.jpg)
A point at the beginning and an arrow at the end. The length of the arrow corresponds to the magnitude of the vector.•The direction the arrow points is the vector direction.Vectors are drawn to scale!!
Vectors can be drawn
![Page 6: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/6.jpg)
6
Example
•The direction of the vector is 55° North of East
•The magnitude of the vector is 2.3.
![Page 7: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/7.jpg)
7
Now You Try
Direction:
Magnitude:
47° North of West
2
![Page 8: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/8.jpg)
8
Try Again
Direction:
Magnitude:
43° East of South
3
![Page 9: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/9.jpg)
9
Try Again
It is also possible to describe this vector's direction as 47 South of East.
Why?
![Page 10: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/10.jpg)
There are two ways of reading a vector’s direction.
By comparing to a cardinal directionBy easting convention
Reading directions
![Page 11: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/11.jpg)
expressed as an angle of rotation of the vector about its tailEx. 40 degrees North of West
(a vector pointing West has been rotated 40 degrees towards the northerly direction)
Ex. 65 degrees East of South(a vector pointing South has been rotated 65 degrees towards the easterly direction)
By comparison
![Page 12: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/12.jpg)
a counterclockwise angle of rotation of the vector about its tail from due East.Ex. 30 degrees, 240 degrees
Easting Convention
![Page 13: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/13.jpg)
There are two ways of adding vectorsgraphicallyAnalytically
Resultant - the vector sum of two or more vectors. It is the result of adding two or more vectors together.
Vector addition
![Page 14: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/14.jpg)
Graphic Addition Graphic Addition
Head-to-Tail MethodHead-to-Tail Method1. Draw the first vector with the proper length and orientation.
2. Draw the second vector with the proper length and orientation originating from the head of the first vector.
3. The resultant vector is the vector originating at the tail of the first vector and terminating at the head of the second vector.
4. Measure the length and orientation angle of the resultant.
![Page 15: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/15.jpg)
Graphic Addition Graphic Addition •Ex. 20 m, 45 deg. + 25 m, 300 deg. + 15 m, 210 deg.SCALE: 1 cm = 5 m
![Page 16: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/16.jpg)
Graphic Addition Graphic Addition •Ex. 20 m, 45 deg. + 25 m, 300 deg. + 15 m, 210 deg.SCALE: 1 cm = 5 m
![Page 17: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/17.jpg)
Graphic Addition Graphic Addition •The order of addition doesn’t matter. The resultant will still have the same magnitude and direction.
![Page 18: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/18.jpg)
Analytically Addition Addition Pythagorean TheoremPythagorean Theorem
This works only if the two vectors are at a right angle.
![Page 19: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/19.jpg)
Analytically Addition Addition Pythagorean TheoremPythagorean Theorem
Ex. Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east. Determine Eric's resulting displacement.
![Page 20: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/20.jpg)
Analytically Addition Addition Pythagorean TheoremPythagorean Theorem
Practice A: 10km North plus 5 km West. What is the resultant vector?
Practice B: 30km West plus 40km South. What is the resultant vector?
![Page 21: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/21.jpg)
Analytically Addition Addition Trigonometry:
Let’s try these togetherLet’s try these togetherBack to Practice A and Practice BPractice A: 10km North plus 5 km West. What is the resultant vector?Practice B: 30km West plus 40km South. What is the resultant vector? Remember: SOH CAH TOA
![Page 22: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/22.jpg)
Analytically Addition Addition Trigonometry
This works only if the two vectors are at a right angle.Remember: SOH CAH TOA
![Page 23: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/23.jpg)
Analytically Addition Addition Trigonometry
Ex. Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east. Determine Eric's resulting displacement.Remember: SOH CAH TOA
![Page 24: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/24.jpg)
Analytically Addition Addition Trigonometry:
Let’s try these togetherLet’s try these togetherBack to Practice A and Practice BPractice A: 10km North plus 5 km West. What is the resultant vector?Practice B: 30km West plus 40km South. What is the resultant vector? Remember: SOH CAH TOA
![Page 25: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/25.jpg)
Analytically Addition Addition
Try this…Try this…Remember: SOH CAH TOAA plane travels from Houston, Texas to Washington D.C., which is 1540km east and 1160Km north of Houston. What is the total displacement of the plane?
Answer:1930 km at 37° north of east
![Page 26: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/26.jpg)
Analytically Addition Addition
Try this…Try this…Remember: SOH CAH TOAA camper travels 4.5km northeast and 4.5km northwest. What is the camper’s total displacement?
Answer:6.4km north
![Page 27: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/27.jpg)
Analytically Addition Addition
Try this…Try this…
Pg 89 Practice A #1-4
![Page 28: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/28.jpg)
28
Resolving Vectors/Expressing Vectors as Ordered Pairs
How can we express this vector as an ordered pair?
Use Trigonometry
These ordered pairs are called the components of the vector.
![Page 29: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/29.jpg)
29
![Page 30: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/30.jpg)
30
A good example:
Express this vector as an ordered pair.
Answer:(42.7, 34.6)
![Page 31: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/31.jpg)
Resolving Vectors
Try this…Try this…Remember: SOH CAH TOAFind the components of the velocity of a helicopter traveling 95km/h at an angle of 35° to the ground.
Answer:6.4km north
![Page 32: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/32.jpg)
Resolving Vectors
Try this…Try this…Remember: SOH CAH TOAFind the components of the velocity of a helicopter traveling 95km/h at an angle of 35° to the ground.
Answer:y = 54km/hx = 78km/h
![Page 33: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/33.jpg)
Resolving Vectors
One more…One more…Remember: SOH CAH TOAAn arrow is shot from a bow at an angle of 25° above the horizontal with an initial speed of 45m/s. Find the horizontal and vertical components of the arrow’s initial velocity.
Answer:41m/s, 19m/s
![Page 34: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/34.jpg)
Resolving Vectors
Try this…Try this…
Pg 92 Practice B #1-4
![Page 35: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/35.jpg)
Resolving VectorsResolving Vectors
What if the vectors aren’t at right angles?Resolving a vector is breaking it down into its x and y components.First, we need a vector.
North39East
s
m47 o
![Page 36: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/36.jpg)
Resolving VectorsResolving Vectors
What if the vectors aren’t at right angles?Resolving a vector is breaking it down into its x and y components.First, we need a vector.
North39East
s
m47 o
![Page 37: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/37.jpg)
39o
E
Let’s draw the vector
s
m47
![Page 38: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/38.jpg)
Continuing
Next we will draw in the component vectors which we are looking for.
![Page 39: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/39.jpg)
Horizontal
Drawing the Components
Vertical
39o
s
m47
![Page 40: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/40.jpg)
Identifying the Sides
Vertical
opphyp
adj
39o
s
m47
Horizontal
![Page 41: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/41.jpg)
What Trig Function will give the Horizontal Component?
opphyp
adj
39o
s
m47
Vertical
Horizontal
coscos adj
hyp
![Page 42: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/42.jpg)
Finding The Horizontal Component
adjm
so 47 39cos
adjm
s36 5.
hyp
adjcos adj
hyp
adjcos coshypdja
hyp
adjcos
![Page 43: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/43.jpg)
opphyp
adj
Finding The Vertical Component
39o
s
m47
Vertical
Horizontal
sinsin opp
hyp
![Page 44: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/44.jpg)
oppm
so 47 39sin
Finding The Vertical Component
oppm
s 29 65.
hyp
oppsin opp
hyp
oppsin sinhypopp
hyp
oppsin
![Page 45: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/45.jpg)
Continuing
The two components are:x:36.6m/sy: 29.65m/s
![Page 46: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/46.jpg)
Resolving Vectors PracticeResolving Vectors PracticeA plane takes off at a 35° ascent with a velocity of 195 km/h. What are the horizontal and vertical components of the velocity?
A child slides down a hill that forms an angle of 37° with the horizontal for a distance of 24.0 m. What are the horizontal and vertical components?
How fast must a car travel to stay beneath an airplane that is moving at 105 km/h at an angle of 33° to the ground (What is the horizontal component?) What is the vertical component of the plane’s velocity?
![Page 47: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/47.jpg)
Resolving Vectors
More practice… More practice…
Pg 92 Practice B #1-4
![Page 48: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/48.jpg)
Analytically Addition Addition
![Page 49: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/49.jpg)
Analytically Addition Addition
![Page 50: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/50.jpg)
Analytically Addition Addition
![Page 51: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/51.jpg)
Analytically Addition Addition
What if the vectors aren’t at right angles?There are four steps.1.We have to resolve the vectors into their components.2.Add all the x components.3.Add all the y components. 4.Find the magnitude and direction of the resultant.
![Page 52: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/52.jpg)
Analytically Addition Addition
A hiker walks 27km from her camp at 35° south of east. The next day, she walks 41km at 65° north of east and discovers a forest ranger’s tower. Find the magnitude and direction of her resultant displacement between the base camp and the tower.
Answer:45km at 29° north of east
![Page 53: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/53.jpg)
Analytically Addition Addition
A camper walks 4.5km at 45° north of east then 4.5 km due south. Find the camper’s total displacement, including direction.
Answer:3.4km at 22° south of east
You try…You try…
![Page 54: 2-D motion](https://reader030.vdocuments.us/reader030/viewer/2022032612/568131f1550346895d98503f/html5/thumbnails/54.jpg)
Resolving Vectors
More practice… More practice…
Pg 94 Practice C #1-4