2-D motion

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.

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.

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.

 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

6 Example The direction of the vector is 55° North of East The magnitude of the vector is 2.3.

7 Now You Try Direction: Magnitude: 47° North of West 2

8 Try Again Direction: Magnitude: 43° East of South 3

9 Try Again It is also possible to describe this vector's direction as 47 South of East. Why?

 There are two ways of reading a vector’s direction.  By comparing to a cardinal direction  By easting convention Reading directions

 expressed as an angle of rotation of the vector about its tail  Ex. 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

 a counterclockwise angle of rotation of the vector about its tail from due East.  Ex. 30 degrees, 240 degrees Easting Convention

 There are two ways of adding vectors  graphically  Analytically  Resultant - the vector sum of two or more vectors. It is the result of adding two or more vectors together. Vector addition

Graphic Addition Head-to-Tail Method 1. 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.

Graphic Addition Ex. 20 m, 45 deg m, 300 deg m, 210 deg.SCALE: 1 cm = 5 m

Graphic Addition Ex. 20 m, 45 deg m, 300 deg m, 210 deg.SCALE: 1 cm = 5 m

Graphic Addition The order of addition doesn’t matter. The resultant will still have the same magnitude and direction.

Addition Analytically Addition Pythagorean Theorem  This works only if the two vectors are at a right angle.

Addition Analytically Addition Pythagorean Theorem  Ex. Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east. Determine Eric's resulting displacement.

Addition Analytically Addition Pythagorean 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?

Addition Analytically Addition Trigonometry: Let’s try these together  Back to Practice A and Practice B  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?  Remember: SOH CAH TOA

Addition Analytically Addition Trigonometry  This works only if the two vectors are at a right angle.  Remember: SOH CAH TOA

Addition Analytically 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

Addition Analytically Addition Trigonometry: Let’s try these together  Back to Practice A and Practice B  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?  Remember: SOH CAH TOA

Addition Analytically Addition Try this…  Remember: SOH CAH TOA  A 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

Addition Analytically Addition Try this…  Remember: SOH CAH TOA  A camper travels 4.5km northeast and 4.5km northwest. What is the camper’s total displacement?  Answer: 6.4km north

Addition Analytically Addition Try this…  Pg 89 Practice A #1-4

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.

29

30 A good example: Express this vector as an ordered pair. Answer: (42.7, 34.6)

Resolving Vectors Try this…  Remember: SOH CAH TOA  Find the components of the velocity of a helicopter traveling 95km/h at an angle of 35° to the ground.  Answer: 6.4km north

Resolving Vectors Try this…  Remember: SOH CAH TOA  Find the components of the velocity of a helicopter traveling 95km/h at an angle of 35° to the ground.  Answer: y = 54km/h x = 78km/h

Resolving Vectors One more…  Remember: SOH CAH TOA  An 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

Resolving Vectors Try this…  Pg 92 Practice B #1-4

Resolving 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.

Resolving 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.

39 o E Let’s draw the vector

Continuing Next we will draw in the component vectors which we are looking for.

Horizontal Drawing the Components Vertical 39 o

Identifying the Sides Vertical opp hyp adj 39 o Horizontal

What Trig Function will give the Horizontal Component? opp hyp adj 39 o Vertical Horizontal

Finding The Horizontal Component

opp hyp adj Finding The Vertical Component 39 o Vertical Horizontal

Finding The Vertical Component

Continuing The two components are: x:36.6m/s y: 29.65m/s

Resolving Vectors Practice  A 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?

Resolving Vectors More practice…  Pg 92 Practice B #1-4

Addition Analytically 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.

Addition Analytically 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

Addition Analytically 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…

Resolving Vectors More practice…  Pg 94 Practice C #1-4