Vectors and Linear Motion. Vector Quantities: Have a magnitude And direction ex: meters, velocity,...

Vectorsand

Linear Motion

Vector Quantities:

Have a magnitude And direction

ex: meters, velocity, acceleration

Scalar Quantities:

Have only a magnitude

ex: temperature, speed, time

Distance

• A length between two points

• Scalar quantity

• Unaffected by direction

Displacement

Distance and direction between 2 positions

Vector quantity

Ex: 5m West

Acceleration

Motion of acceleration

Motion of acceleration

Dimensional Analysis

Dimensional Analysis

 Step 1: Multiply by 1

Dimensional Analysis

 Step 2: Cancel units

Dimensional Analysis

 Step 3: Repeat

Dimensional Analysis

 Step 3: Repeat

Dimensional Analysis

 Step 4: Solve

Unit 2:

Acceleration and Freefall

Gravity

• Pulls on all objects with the same acceleration (neglecting air resistance)

• Earth’s gravitational field decreases as you travel above Earth’s surface

• g = -9.8m/s2

An object is dropped from an airplane and falls freely for 20s before hitting the ground.

A) How far did the object fall?

B) What was it’s velocity after 20s?

Objects Thrown Straight Up

• For objects launched upward, the time to reach the top of their flight is half the total flight time

• Vy @ top is 0m/s

• a=-9.8m/s2 always

• vi = -vf

A physics student throws a tennis ball into the air. It takes 6s for the tennis ball to land.

A) How high did the tennis ball go?

B) What was it’s initial velocity?

Velocity and Acceleration Graphs

Velocity and Distance Graphs

Distance

Velocity

Projectile Motion

Independence of Dimensions

Vertical and Horizontal motions are treated separately

We calculate one at a time

Use the X-Y chart to show your givens

The X-Y Chart

X Y

a

vi

vf

t

d

Horizontal Values

vi = vf in the x direction

a is always 0m/s2 in the X direction

time is the same as the vertical time

Vertical Values

a is always -9.81 m/s2 in the Y direction

Find time or distance using:

d=vit+(1/2)at2

A ball is rolling 5m/s and rolls off of a 10m ledge. How far from the ledge does the ball land?

5 m/s

10 m

A ball is rolling 5m/s and rolls off of a 10m ledge. How far from the ledge does the ball land?

5 m/s

10 m

A ball is rolling 5m/s and rolls off of a 10m ledge. How far from the ledge does the ball land?

5 m/s

10 m

Velocity VectorsVelocity vectors

A ball is rolling 5m/s and rolls off of a 10m ledge. How far from the ledge does the ball land?

5 m/s

10 m

Acceleration Vectors

Acceleration vectors (Notice they don’t change)

A ball is rolling 5m/s and rolls off of a 10m ledge. How far from the ledge does the ball land?

5 m/s

10 m

X Y

a

vf

vi

t

d

A ball is rolling 5m/s and rolls off of a 10m ledge. How far from the ledge does the ball land?

5 m/s

10 m

X Y

a

vf

vi 5

t

d 10

A ball is rolling 5m/s and rolls off of a 10m ledge. How far from the ledge does the ball land?

5 m/s

10 m

X Y

a 0 -9.81

vf 5

vi 5 0

t

d ? 10

A ball is rolling 5m/s and rolls off of a 10m ledge. How far from the ledge does the ball land?

5 m/s

10 m

X Y

a 0 -9.81

vf 5

vi 5 0

t 1.43 1.43

d ? 10

A ball is rolling 5m/s and rolls off of a 10m ledge. How far from the ledge does the ball land?

5 m/s

10 m

X Y

a 0 -9.81

vf 5

vi 5 0

t 1.43 1.43

d 7.15 10

Try this one on your own

A careless driver left a car in neutral near the edge of a 15m high cliff. The car rolled at a velocity of 3m/s as it rolled off the cliff.

A) How long did it take the car to reach the bottom?

B) How far from the cliff did it land?

X Y

a

vf

vi 3

t

d 15

Givens:

X Y

a 0 -9.8

vf 3

vi 3 0

t

d 15

We also know:

X Y

a 0 -9.8

vf 3

vi 3 0

t 1.75 1.75

d 5.25 15

Objects Launched at An Angle

Break the velocity down into x and y components

30°

10m/s10sin(30°)

10cos(30°)

SO

HCA

H TOA

Sin(θ) = Opposite Hypotenuse

Cos(θ) = Adjacent

Hypotenuse

Tan(θ) = Opposite Adjacent

θ

Opposite

Adjacent

Objects Launched at An Angle

Break the velocity down into x and y components

30°

10m/s10sin(30°)

10cos(30°)

Objects Launched at An Angle

Break the velocity down into x and y components

30°

10m/svy = 5m/s

Vx = 8.66m/s

Start filling in your X-Y chart

Solve for one dimension at a time

X Y

a

vf

vi

t

d

X Y

a

vf

vi 8.66 5

t

d

X Y

a 0 -9.81

vf 8.66 -5

vi 8.66 5

t

d

X Y

a 0 -9.81

vf 8.66 -5

vi 8.66 5

t 1.01 1.01

d 8.57 10.05

What happens as θ increases?

10m/s

10°

What happens as θ increases?

10m/s

10°

10m/s sin(10°)

10m/s cos(10°)

What happens as θ increases?

10m/s

10°

1.74 m/s

9.84 m/s

What happens as θ increases?

10m/s

55°

10m/s sin(55°)

10m/s cos(55°)

What happens as θ increases?

10m/s

55°

8.19 m/s

5.74 m/s

Range

Horizontal Distance a projectile travels before landing

For a given velocity, 45° produces the greatest range

Range

Air Resistance

No Air Resistance

Air Resistance

With Air Resistance

No Air Resistance

Forces And Newton’s Laws

A Force is a push or a pull on an object

Units- Newtons (N)

Forces only exist as a result of an interaction

Contact Force- Results from 2 objects touching each other

Field Force- A push or a pull between 2 bodies that aren’t touching.

What field force have we discussed so far in class? Can you think of any others?

FN

Fg

For an object resting on a level surface FN = -Fg

Concurrent Forces

Two or more forces acting on the same object at the same time are considered to be concurrent with each other.

They are sometimes said to be acting concurrently

When you will see the term “Concurrent”:

The magnitude of the resultant of two concurrent forces is a minimum when the angle between them is:

A) 0° B) 45° C) 90° D) 180°

When you will see the term “Concurrent”:

The magnitude of the resultant of two concurrent forces is a minimum when the angle between them is:

A) 0° B) 45° C) 90° D) 180°

Resultant

Forces are vector quantities. They can be expressed and added using arrows just like distance and velocity vectors. We call this sum the resultant

2N + 2N = 4N

Net Force

FNet

Sum or “Net” of all forces acting on an object

FNet = F1 + F2 + F3…Not in reference table

Equilibrium

A state where FNet is 0N

A force that causes equilibrium is called an equilibrant

An equilibrant is equal in magnitude to, but opposite in direction of a resultant (or FNet)

In other words:

What force would we need to cancel out or neutralize the forces we have now?

Components

• We can find the components of forces the same way we found components of velocities and distances

FFy

Fx

Fx = F cos(θ)

Fy = F sin(θ)θ

Free Body Diagrams (FBD’s)

• FBD’s are used to show the forces acting on an object.

• Start with a square and draw all forces originating from the center of the square pointing outward

**Ignore MC questions that show it differently**

What does the FBD look like for an object that weighs 1 N and is being pulled on to the right with a force of 1N, but being held still by a force of friction of 1N?

2 N

1 N 2 N

1 N

Find FNet:

2 N

1 N 2 N

1 N

Step 1:Find Fy and Fx:

1 N

1 N

1 N

1 N

FNet

Step 2: Add your vertical and horizontal vectors

1 N

1 N

1 N

1 N

1.4N

Step 3: Solve

1 N

1 N

1 N

1 N1.4N

Newton’s First Law

“An object at rest will stay at rest and an object in motion will stay in motion unless acted on by an unbalanced force.”

Inertia- The resistance of an object to a change in motion

Inertia is directly proportional to mass

Which Object Has More Inertia?

Which Object Has More Inertia?

A B

20 m/s Right10 m/s Right

Newton’s Second Law

• The acceleration of an object is directly proportional to the net force acting on it and indirectly proportional to it’s mass.

Think of it this way:• Bigger forces cause bigger accelerations

• Bigger masses cause smaller accelerations

Newton’s Second Law

• You can express Newton’s 2nd Law as an equation:

Mass vs Weight

• Mass- Amount of matter in an object– Units- kg– Unchanged by gravitational field

• Weight- Force of pull on an object caused by gravity– Units- N– Depends on gravitational field strength

Weight (Fg)

• Always directed towards Earth (down)

• Force of Gravity

• Fg = mg g=Fg/m Units: m/s2 = N / Kg

• g = 9.8m/s2

Newton’s Third Law

“For every action there is an equal and opposite reaction”

If a person weighing 700N is standing in an elevator that’s accelerating up, what force does the elevator apply on the person?

Fg

FNet =

a = + 2 m/s2

m = 70kg

FN=

Ff = µFN

Solve this equation for µ and predict what you think the units for µ will be

µ = Ff / FN

µ is unit-less

Inclined planes

• We will call this force FII

FN

Fg

FII

Inclined planes

• FII is a component of the object’s weight along

with F˔FN

Fg FII

F˔

Calculating FII and F˔Sin(θ) = FII / Fg FII = Fg sin(θ)

Cos(θ) = F˔ / Fg F˔ = Fgcos(θ)

What do FII and F˔ mean?

F˔ has the same magnitude as the normal force

FN

F˔θ

What do FII and F˔ mean?FII is the component of the weight that is parallel to the plane

FII

θ

FII = -Ff when FNet = 0N

FII

θ

Ff