Algebra Skills & Proportional Reasoning. Rearranging One of the tasks that physics requires is...

Algebra Skills&

Proportional Reasoning

Rearranging

• One of the tasks that physics requires is being able to rearrange equations.

• Remember:The reason for rearranging is to isolate the variable that you are looking for.

• Basic Rule: What you do to one side of the equation, you must do to the other side also.

Example: v = d/t

• Multiple both sides by t. The t’s on the right hand side cancel leaving d.

• Therefore, d = vt• Solve for t• Again, multiply both

sides by t and divide both sides by v.

• Therefore, t = d/v

Solve for d

Example: d = ½ g t2

• Multiple both sides by 2

• 2d = gt2

• Divide both sides by g

• 2d/g = t2

• Square root both sides.

• T = √(2d/g)

Solve for time.

Example: E = mgh + ½ mv2

• E = m(gh + ½ v2)

• m = E/(gh + ½ v2)

Solve for m

Example 2: E = mgh + ½ mv2

• E – ½ mv2 = mgh

• h = (E – ½ mv2 )/mg

Solve for h

Example 3: E = mgh + ½ mv2

• E – mgh = ½ mv2

• 2(E – mgh ) = mv2

• 2(E – mgh )/m = v2

• v = √[2(E – mgh )/m]

Solve for v

Example 4: Solve by Substitution

2x + 8y = 1 2x + 8y = 1

2(2y) + 8y = 12(2y) + 8y = 1

4y + 8y = 1 4y + 8y = 1

12y=1 12y=1

y = y = 11//1212

Since x = 2y, you can insert 2y wherever x occurs.

Unsolvable on its own….Unsolvable on its own….

But if two equations are known…But if two equations are known…x = 2yx = 2y

Solve for x and ySolve for x and y

x = 2y x = 2y

y = y = 11//1212

x = 2(x = 2(11//1212))

y = y = 11//66

Now you can solve for x:

Describing MotionDescribing Motion

• Motion can be described using words.Motion can be described using words.• Motion can be described using diagrams.Motion can be described using diagrams.• Motion can be described using equations.Motion can be described using equations.• Motion can be described using graphs.Motion can be described using graphs.

Vocabulary:Vocabulary:

Scalar vs. VectorScalar vs. Vector

A scalar quantity has A scalar quantity has magnitudemagnitude only. only.– Examples: distance, temperatureExamples: distance, temperature

A vector quantity has A vector quantity has magnitudemagnitude and and directiondirection..– Examples: force, acceleration.Examples: force, acceleration.

Symbols for vector quantities are written in Symbols for vector quantities are written in boldbold or with or with an arrow above them:an arrow above them:

a

Beginning Question:Beginning Question:

A teacher walks 5.0m north of his desk, and then turns around and walks 6.0m south. How far has the teacher gone?

11.0m? … or 1.0m?

“How far has he gone” is not clear enough.We need to distinguish between

Distance vs. Displacement

Vocabulary:Vocabulary:

Distance:Distance: How far an object has traveled How far an object has traveled

A A scalarscalar quantity: has magnitude only. quantity: has magnitude only.

Displacement:Displacement: Change in position Change in position

A A vectorvector quantity: has magnitude and direction. quantity: has magnitude and direction.

DisplacementDisplacement

if xxx

01 ddd

01 yyy

01 rrr

oror

oror

oror

most common most common notationnotation

• Motion can be described using words.Motion can be described using words.• Motion can be described using diagrams.Motion can be described using diagrams.• Motion can be described using equations.Motion can be described using equations.• Motion can be described using graphs.Motion can be described using graphs.

……but always depends on a FRAME OF REFERENCE.but always depends on a FRAME OF REFERENCE.

……………………………………..assign a coordinate system...assign a coordinate system.

http://http://acme.highpoint.edu/~atitus/physlets/index.htmlacme.highpoint.edu/~atitus/physlets/index.html

Sample problem #1Sample problem #1

A teacher walks 5.0m north of his desk. What is the teacher’s displacement?

The desk is at xi = 0.0m

The teacher’s final position, xf = 5.0m north

The displacement, x = xf – xi

x = 5.0m – 0.0mx = 5.0m

Sample problem #2Sample problem #2

Find the displacement of the gecko.Find the displacement of the gecko.

Sample problem #3Sample problem #3

Find the Find the displacement of displacement of the gecko.the gecko.

If a displacement is written without a direction stated, assume it is in the x – direction.

eg: d = 54.9 m

Sample problem #4Sample problem #4

A teacher walks 5.0m north of his desk. He then walks 6.0m south. What is the teacher’s displacement in reference to his desk? What is the total distance the teacher has walked?

Forget about the formula and think this through.If north is positive and south is negative, thenx = 5.0m – 6.0m = -1.0m

or 1.0m south of the desk

Sample problem #4Sample problem #4

A teacher walks 5.0m north of his desk. He then walks 6.0m south. What is the teacher’s displacement in reference to his desk? What is the total distance the teacher has walked?

Distance = total amount traveled

Distance = 5.0m + 6.0m = 11.0m walked

Notice how the direction is not considered.

Distance vs. DisplacementDistance vs. Displacement

5.0m North

Distance vs. DisplacementDistance vs. Displacement

6.0m South

Distance vs. DisplacementDistance vs. Displacement

6.0m South

5.0m North

1.0m South

The displacement tells you where the teacher ended up.The distance tells you the total length of his journey.

Changing position over timeChanging position over time

http://http://acme.highpoint.edu/~atitus/physlets/index.htmlacme.highpoint.edu/~atitus/physlets/index.html http://http://acme.highpoint.edu/~atitus/physlets/index.htmlacme.highpoint.edu/~atitus/physlets/index.html

How fast is an object moving?How fast is an object moving?

Average Velocity:Average Velocity: v v – displacement : time ratiodisplacement : time ratio

– A A vectorvector quantity: has magnitude and direction. quantity: has magnitude and direction.

– total displacement : total time elapsedtotal displacement : total time elapsed

Average Speed:Average Speed: – distance : time ratiodistance : time ratio

– A A scalarscalar quantity: has magnitude and direction. quantity: has magnitude and direction.

– total distance : total time elapsed total distance : total time elapsed

v

s

Average VelocityAverage Velocity

01

01

tt

dd

t

dv

Velocity vs. SpeedVelocity vs. Speed

Velocity is a vector

Velocity = displacement

time

Speed is a scalar (direction does not matter)

Speed = distance

time

Speed and VelocitySpeed and Velocity

Can an object have a velocity that is changing Can an object have a velocity that is changing while the speed remains the same?while the speed remains the same?

Can an object have a speed that is changing Can an object have a speed that is changing while the velocity remains the same?while the velocity remains the same?

Instantaneous VelocityInstantaneous Velocity

Instantaneous velocity is the velocity at a Instantaneous velocity is the velocity at a given point in time.given point in time.

Example: Speedometer, Radar gunExample: Speedometer, Radar gun

Constant VelocityConstant Velocity

Constant velocity is when an objects Constant velocity is when an objects velocity remains the same for a given velocity remains the same for a given amount of time.amount of time.

Sample Problem #5Sample Problem #5

Suzy Physics Student lives 5.0miles south of school.Suzy Physics Student lives 5.0miles south of school.

If she takes 2.0 hours to get to school, what is If she takes 2.0 hours to get to school, what is Suzy’s average velocity?Suzy’s average velocity?

Sample Problem #6Sample Problem #6

Suzy Physics Student walks 5.0miles South to school.Suzy Physics Student walks 5.0miles South to school.

She takes 2.0 hours to get to school, realizes she is She takes 2.0 hours to get to school, realizes she is hungry and decides to walk for 1.0 hour to go 2.0 hungry and decides to walk for 1.0 hour to go 2.0 miles North to IHOP.miles North to IHOP.

What is Suzy’s average velocity for the whole trip? What is Suzy’s average velocity for the whole trip?

What is Suzy’s average speed for the whole trip?What is Suzy’s average speed for the whole trip?

Sample Problem #7Sample Problem #7

Practice problemsPractice problems

Practice problemsPractice problems

Practice problemsPractice problems

Practice problemsPractice problems

Practice problemsPractice problems

Linear Relationships: y = k xLinear Relationships: y = k x

0

10

20

30

40

50

10 20 30 40

Volume(mL)

Mass

(g)

Slope

m=(40-8)/(50-10)

m=32/40

m=0.8 g/cm3

Interpolation

vs.

Extrapolation

Graphing MotionGraphing Motion

• Motion can be described using words.Motion can be described using words.• Motion can be described using diagrams.Motion can be described using diagrams.• Motion can be described using equations.Motion can be described using equations.• Motion can be described using graphs.Motion can be described using graphs.

Graphing MotionGraphing Motion

Graphs of Position vs. Time Graphs of Position vs. Time

1.1. Calculate the displacementCalculate the displacement

2.2. Calculate the velocityCalculate the velocity

3.3. Describe forward and reverse motionDescribe forward and reverse motion

4.4. Describe an object staying stillDescribe an object staying still

Calculating displacementCalculating displacement

A position vs. time graph lets you calculate the A position vs. time graph lets you calculate the displacement between any two moments:displacement between any two moments:

x, P

osit

ion,

(m

)

t, Time, (s)

Calculating displacementCalculating displacement

Find the displacement between A and BFind the displacement between A and B

x, P

osit

ion,

(m

)

t, Time, (s)A (0,0)

B (1,5) Use x = xf –xi

Where xf = 5m

and xi = 0m

x = xf – xi

x = 5m – 0m

x = 5m

Calculating displacementCalculating displacement

Find the displacement between C and DFind the displacement between C and D

x, P

osit

ion,

(m

)

t, Time, (s)

C (4,8)

D (8,3)

Use x = xf –xi

Where xf = 3m

and xi = 8m

x = xf – xi

x = 3m – 8m

x = -5m

Calculating displacementCalculating displacementFind the displacement from when the object was Find the displacement from when the object was moving for 2s to when it had been moving for 9s.moving for 2s to when it had been moving for 9s.

x, P

osit

ion,

(m

)

t, Time, (s)

Use x = xf –xi

Where xf = 6m

and xi = 5m

x = xf – xi

x = 6m – 5m

x = 1m2s 9s

5m

6m

Position vs. Time GraphPosition vs. Time Graph

What are the velocities? What are the velocities?

What are the velocities?What are the velocities?

Average Velocity on an x-t graphAverage Velocity on an x-t graph

vvavav vs v vs vinstinst on an x-t graph on an x-t graph

x-t graph vs motion of a particlex-t graph vs motion of a particle

AccelerationAcceleration

• Motion can be described using words.Motion can be described using words.• Motion can be described using diagrams.Motion can be described using diagrams.• Motion can be described using equations.Motion can be described using equations.• Motion can be described using graphs.Motion can be described using graphs.

TermsTerms

Acceleration vs. AccelerationAcceleration vs. Acceleration

Acceleration:Acceleration: The rate at which The rate at which velocity velocity changes changes (vector)(vector)

Acceleration:Acceleration: The rate at which The rate at which speedspeed changes changes (scalar)(scalar)

Symbol: a Symbol: a or or aa

SI Unit:SI Unit: meters per second per second or meters per second per second or meters per second squared, m/s/s or m/smeters per second squared, m/s/s or m/s22

Average AccelerationAverage Acceleration

Velocity vs. Time GraphVelocity vs. Time Graph

The The slopeslope of a of a v vs. tv vs. t graph is the graph is the accelerationacceleration

The The areaarea between the curve and the between the curve and the horizontal axis of a horizontal axis of a v vs. tv vs. t graph is graph is the displacementthe displacement

Instantaneous Acceleration vs Average Instantaneous Acceleration vs Average Acceleration from a v-t graphAcceleration from a v-t graph

v-t graph vs motion of a particle v-t graph vs motion of a particle

x-t graph vs motion of x-t graph vs motion of a particle with a particle with accelerationacceleration

What are the accelerations and What are the accelerations and displacements?displacements?

What are the accelerations and What are the accelerations and displacements?displacements?

Acceleration vs. Time GraphAcceleration vs. Time Graph

The The slopeslope means means NOTHINGNOTHING

The The areaarea between the curve and the horizontal between the curve and the horizontal axis is the change in velocityaxis is the change in velocity

a

t-10 m/s/s

10 s

ImportantImportant

AccelerationAcceleration tells us how fast tells us how fast velocityvelocity changeschanges

VelocityVelocity tells us how fast tells us how fast positionposition changeschanges

Kinematics EquationsKinematics Equations(accelerated motion)(accelerated motion)

Falling Bodies, thrown up objects, and Falling Bodies, thrown up objects, and the y-directionthe y-direction

All things fall at the same rate All things fall at the same rate (neglecting air resistance)(neglecting air resistance)

On earth that rate is 9.80 m/sOn earth that rate is 9.80 m/s22

That rate is an accelerationThat rate is an acceleration

The name of that acceleration is The name of that acceleration is GravityGravity

Object moving in y-dirObject moving in y-dir