Lesson 3-1Review of Vectors

Scalars and Vectors

Recall a scalar does not have a direction A vector has BOTH magnitude and direction Vectors can be added graphically

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8

9

4

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Similar Quantities

When adding vectors, the units must match It would be meaningless to add a force vector to a

velocity vector They are essentially apples and oranges

When vectors do have the same units, we may add or subtract the vectors

Example

A student is walking to school. First, the student walks 350m to a friend’s house. The two then both walk 740 m to school.

The method to add the vectors is called the tail to tip method. The vector we find is called the resultant vector.

Moving Vectors

Vectors can be moved parallel to each other Does not matter where the vectors are, as long as

they are addable, tail to tip Example

Push a toy car across a moving sidewalk Say the sidewalk is moving at 1.5 m/s The car is pushed .8 m/s

Vector Addition and Subtraction Vector addition is commutative

The order the vectors are added does not matter To subtract a vector, simply add the opposite

Multiplying and Dividing Vectors Multiplying or Dividing vectors by scalars

results in vectors Lets say we have the velocity of a race car

We want to examine the properties of the car when it is traveling twice as fast

If vi is v, what is twice vi?

What is half of vi

What would be the new v if the car drove twice as fast in the opposite direction?

Lesson 3-2Vector Operations

Coordinate Systems

Up to this point, we have only needed one dimension to study our situations What if we wanted to study a ball being thrown at

45o above the ground? That path of motion does not fit any of our current

axis We will have to use a combination of the two axis

Note: Orientation of the axis is up to you.

Determining the Resultant

Trigonometry is very useful to find the resultant vector. The Pythagorean Theorem Think of a tourist in Egypt walking up the side of a

pyramid Are they walking vertical? Are they walking horizontal? It is a combination of the two motions that produces one

final motion, somewhere between horizontal and vertical

Resultant

The resultant of two vectors is also a vector That means the resultant must have:

Magnitude Direction

It is not enough to say the magnitude of the resultant, it must have direction also.

We will use the trig functions of sine, cosine or tangent to find the direction

Resolving Vectors

Any vector may be broken into x and y components That is to say any vector may be RESOLVED into

its component vectors A horizontal vector has a 0 y component A vertical vector has a 0 x component A vector at 45o has equal x and y vectors

Examples

Page 92, Film Crew

Pg 93

Non-perpendicular Vectors

Until now, all of our vectors have been perpendicular to each other Things in real life are much, much less rigid

Lets say a plane travels 50 km at an angle of 35o, then levels out and climbs at 10o for 220 km These vectors are not perpendicular, we cannot

use the Pythagorean Theorem, yet Resolve the vectors

Lesson 3-3Projectile Motion

Two Dimensional Motion

In the last section, we resolved vectors into x and y components.

We will apply the same ideas to something thrown or flying in the air

Think of a long jumper When she approaches her jump, she has only an

x component When she jumps, she has both x and y

components

Analyze Projectile Motion

We can break the motion into the two component vectors and apply the kinematic equations one dimension at a time

Any object thrown or launched into the air that is subject to gravity is said to have projectile motion

Examples?

Projectile Path

Projectiles follow a path called a parabola A common mistake is to assume projectiles fall

straight down Since there is vxi, there must be continuous

horizontal motion

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Vx

Vy

Projectile Path

Neglecting air resistance, is there anything to stop the projectile in the horizontal direction?

Velocity in the horizontal direction is constant In real life, horizontal velocity is not constant, but for our

purposes we will assume uniform, constant velocity

V

Vx

Vy

Projectile Path

Projectile motion is simply free fall with horizontal velocity

If two similar objects fall to the ground from the same height, one straight down while the other is thrown out to the side, which hits first?

It is very important to realize motion in the x direction is completely independent of motion in the y direction

Summary

A projectile has horizontal velocity until the object stops (hits the ground)

A projectile will have a vertical velocity that is ever changing due to gravity, until the projectile stops (hits the ground)

What is the only factor that is consistent in the x AND y directions? Time

Projectile Path

V

Vx

Vy

Sample Pg 101, Practice Pg 102

Objects Launched at an Angle When an object is launched at an angle, the

object has both horizontal and vertical velocity components

This is similar to the motion of an object thrown straight up with an initial vertical velocity

Example pg 103, Practice pg 104

Lesson 3-4Relative Motion

Frames of Reference

Velocities are different in different frames of reference You are in a train traveling at 40 km/h

Relative to the train, how fast are you moving? Someone outside sees the train pass, how fast do they

see you moving?

The velocities are different because the reference frames were different You – Train, Outside observer - Earth

Examples

You are driving on the interstate at 80 km/h and a car passes you at 90 km/h How fast does it seem the passing car is moving

to you? To someone on the side of the road? A semi-truck driving west at 85 km/h passes

a car on the other side of the road, driving east at 75 km/h. To the trucker, how fast is the car moving?

Examples

A person standing on top of a train traveling at 20 km/h. They throw a baseball. How fast does it look like the ball is moving to a person standing on the ground when:

The ball is thrown 10 km/h forward

The ball is thrown 40 km/h backward

The ball is thrown 20 km/h backward

The ball is thrown straight up

Example pg 108, Practice pg 109