Chapter 3 Intro to Vectors Notes

Section 1-3 Notes

Scalars & Vectors

Scalar – a quantity that has magnitude but no direction Units must be expressed Ex: mass, volume, speed, time, distance, counting numbers

Vector – a quantity that has magnitude and direction Units & direction must be expressed Ex: velocity, acceleration, force, displacement

Module 5 - 2

Vector Addition – One DimensionIf an object’s motion involves more than one vector, these can be added to find a net description of the motion

A person walks 8 km East and then 6 km East.

Displacement = 14 km East

A person walks 8 km East and then 6 km West.

Displacement = 2 km

Adding Vectors Graphically When vectors are added the answer is called

the resultant To add vectors graphically, they must be

drawn tail to tip Ex: A bus travels north 10.0km and west

10.0km. What is the resultant displacement?

Displacement = 14.1km to the northwest

10.0km

10.0km

R

Ex: A bus travels north 10.0km and west 10.0km. What is the resultant displacement?

D= 14.1km, 135° from the horizontal

Module 5 - 5

Graphical Method of Vector AdditionTail to Tip Method

1V

2V

3V

RV

1V

2V

3V

Adding Vectors Graphically – a little more complicated with graph paper

A student walks 150m NE to his friend’s house. The student then walks another 800m NW to get to the bus stop. What is the displacement?

150m

800m

R

Properties of Vectors

Vectors can be moved parallel to themselves in a diagram to get them tail to tip

Ex: A dog swims across a flowing river.

End

Start

River velocity

Dog velocity

End River velocity

Vectors can be added in any order. Ex: A bus travels from city A to city B.

OR

To subtract a vector, add its opposite. Mathematically this means change from positive to negative or vice versa. Graphically it means switch the direction.

Ex: An airplane traveling at 50m/s is slowed by a wind of 5m/s.

50m/s

-5m/s45m/s Fp = 50NFfr = 10N

Fnet = 40N to the right

Vectors that are multiplied by scalars result in vectors

Ex: A customer tells a cab driver to drive twice as fast

2 x 12m/s = 24m/s

Scalar Vector

Practice Problems – use graph paper across from your Cornell notes1) A boat moves at 0.80m/s across a river that is flowing at

1.5m/s. What is the resultant velocity of the boat? (solve graphically and algebraically)

2) A telephone pole support cable is in the way of some construction workers. In order for the work to proceed, the cable must be moved 2 meters closer to the pole. If the pole is 10 meters tall and the cable is currently fastened to the ground 8 meters from the pole, how much will the workers need to cut off from the cable when they move it?

#2 Solution: diagram the problem

Solution

The original cable length is

or 12.8m.

By moving the cable 2 meters closer to the pole, we shorten the overall length of the cable to

Or 11.7m. Therefore, 1.1m must be cut off the cable.

Ch 3 Homework #1

Questions: 1,5,8,9 on pg 67Problems: 1,5,8,9 on pg. 68

Due October 26

Ch 3 Vector Quiz on Oct 26

Chapter 3-Resolving Vectors

Section 5,6,7 Notes

Pythagorean Theorem to Find a Resultant (vectors are perpendicular)

Remember: c2 = a2 + b2 (c = hypotenuse)

c is the resultant vector

Use the tangent function to find the direction Tanθ = opp/adj or θ = tan-1(opp/adj)

Example: A runner runs 8.5km to the east and then runs 2.5km to the north. What is the displacement?

8.5km

2.5kmR

θ

Resolving Vectors into Components

Components – the horizontal (x) or vertical (y) parts of a vector

The x and y components of a vector make a right triangle with the vector

Horizontal component

Vertical component

Vector

We can now describe motion in terms of the horizontal motion and vertical motion

Ex: An airplane takes off at an angle of 25° and a speed of 185mi/h. It’s ground speed is 168mi/h. It climbs up at 78mi/h.

168mi/h

78mi/h185mi/h

25°

Module 5 - 13

Signs of Componentsy

x

y

x

R

R

y

x

R

R

y

x

R

R

y

x

R

R

Properties of Right Triangles

SOH CAH TOA sinθ = opp/hyp cosθ = adj/hyp tanθ = opp/adj θ

hyp

adj

opp

SOH CAH TOA Use the cosine function to get the x component Use the sine function to get the y component

168mi/h

78mi/h

185mi/h

25°

Hyp = 185mi/h

Adj = 168mi/h

25°

Opp = 78mi/h

Note: coordinate system has been changed so all is positive

Formulas for Resolving Components

Vx = V(cosθ) Vy = V(sinθ)

Vx = 185(cos25°) Vy = 185(sin25°)

Vx = 168 Vy = 78

185mi/h

25°

Steps for Adding Vectors that are not Perpendicular

We cannot use the pythagorean theorem, so we must 1) After sketching, resolve each vector into its x and y components

Rθθ

V1

V2

V1y

V1x

V2x

V2y

θ

θ

θ1

θ2

2)Add the x components and add the y components

3)Use the Pythagorean theorem to find the magnitude of the resultant

VR2 = Vx2 + Vy2

4)Find the direction of the resultant θ = tan-1(Vy/Vx)

5)Does my answer make sense?

Rθθ

Practice Problems3-9. A man in a rowboat is trying to cross a river that flows due west

with a strong current. The man starts on the south bank and is trying to reach the north bank directly north from his starting point. The boat must a)head due north, b)head due west, c)head in a northwesterly direction, d)head in a northeasterly direction. Use a sketch to justify your answer.

7th edition (3-10,2,3)

3-1. A rural mail carrier leaves the post office and drives 22.0km in a northerly direction to the next town. She then drives in a direction 60.0° south of east for 47.0km to another town. What is her displacement from the post office?

7th edition (3-2,3)

Ch 3 Homework #2

QuestionsProblems

Ch 3 Test on

Physics Chapter 3 Projectile Motion

Section 3 Notes

Projectile Motion- The two dimensional motion of an object that is thrown or

launched into the air.

Module 7 - 1

A projectile is an object moving horizontally under the influence of Earth's gravity; its path is a parabola.

waterslide (1min):https://www.youtube.com/watch?v=W46UMzFEU24Mythbusters (5min):https://www.youtube.com/watch?v=iHu6LVg-0Hs

Concept Facts about Projectile Motion

•Projectile motion is free fall with horizontal velocity (Neglect air resistance)

•Consider motion only after release and before it hits

•Analyze the vertical and horizontal components separately

•a=0 in the horizontal, so vx = constant

•ay = -g = -9.8m/s/s

•Object projected horizontally will reach the ground at the same time as one dropped vertically

•If the ball returns to the y = 0 point, then v = v0.

•Range (x displacement) is determined by time it takes for projectile to return to ground or other end point

Formulas for Projectile Motion

Quadratic Formula:

ax2+bx+c = 0

x = -b± b2-4ac 2a

00 xvv tvxx x00

tgvv yy 0

)(2 02

02 yygvv yy

200 2

1tgtvyy y

Horizontal Motion Vertical Motion

Solving Projectile Motion Problems1. Draw a picture and select a coordinate

system.

2. Write down all the given information.

3. Resolve any vectors (velocity) into x and y components.

4. Choose the formula you need.

5. Solve the problem.

Ex: James thought it was the perfect murder plot: push his wife off a cliff and claim she fell. Ahh, but physics won, and James is in jail. James pushed his wife straight out with a velocity of 0.559m/s. She fell 182m. How far from the cliff did she land?

Practice Problems3-5. A child sits upright in a wagon which is moving to the right at

constant speed. The child extends her hand and throws an apple straight upward while the wagon continues to move forward at constant speed. If air resistance is neglected, will the apple land a) behind the wagon, b)in the wagon, or c) in front of the wagon.

3-6. A boy on a small hill aims his water-balloon slingshot horizontally, straight at a second boy hanging from a tree branch a distance d away. The boy in the tree is the same height up in the air as the water balloon sling shot on the hill. At the instant the water balloon is released, the second boy lets go and falls from the tree, hoping to avoid being hit. Use a diagram and 2-3 sentence explanation to explain that he made the wrong move.

7th edition (3-6,7)

Practice Problems3-3. A movie stunt driver on a motorcycle speeds horizontally off a

50.0m high cliff. How fast must the motorcycle leave the cliff-top if it is to land on level ground below, 90.0m from the base of the cliff where the cameras are.

3-4. A football is kicked at an angle θ=37.0° with a velocity of 20.0m/s. Calculate a) the maximum height, b) the time of travel before the football hits the ground, c) how far away it hits the ground, d)the velocity vector at its maximum height, and e) the acceleration vector at maximum height. Assume the ball leaves the foot at level ground and ignore air resistance and rotation of the ball.

7th edition (3-4,5)

Practice Problems3-8. Suppose the football in example 3-4 (θ=37.0° with a velocity of

20.0m/s) was a punt and left the punter’s foot at a height of 1.00m above the ground. How far did the football travel before hitting the ground? Set x0 = 0, y0 = 0.

7th edition (3-9)

Requires Quadratic

Old Practice Problems1. People in movies often jump from buildings into

pools. If a person jumps from the 10th floor (30.0m) to a pool that is 5.0m from the building, with what initial speed must the person jump?

Practice #102. A golfer practices driving balls off a cliff into the water

below. The cliff is 15m from the water. If the ball is launched at 51m/s at an angle of 15°, how far does the ball travel horizontally before hitting then water?

Practice #103. A missile is shot at a 60.0° angle and a speed of

55.0m/s. It is trying to reach an aircraft that is flying at 450m. Will the missile be able to reach its target? If not, what should be the initial velocity to reach its target?