# Algebra Skills & Proportional Reasoning

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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 Solve for d
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

Example: d = ½ g t2 Solve for time. Multiple both sides by 2 2d = gt2
Divide both sides by g 2d/g = t2 Square root both sides. T = √(2d/g)

Example: E = mgh + ½ mv2 Solve for m E = m(gh + ½ v2)
m = E/(gh + ½ v2)

Example 2: E = mgh + ½ mv2 Solve for h E – ½ mv2 = mgh

Example 3: E = mgh + ½ mv2 Solve for v E – mgh = ½ mv2

Example 4: Solve by Substitution
2x + 8y = 1 Unsolvable on its own…. x = 2y But if two equations are known… Solve for x and y Since x = 2y, you can insert 2y wherever x occurs. Now you can solve for x: 2(2y) + 8y = 1 4y + 8y = 1 12y=1 y = 1/12 x = 2y y = 1/12 x = 2(1/12) y = 1/6

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

Vocabulary: Scalar vs. Vector
A scalar quantity has magnitude only. Examples: distance, temperature A vector quantity has magnitude and direction. Examples: force, acceleration. Symbols for vector quantities are written in bold or with an arrow above them:

Distance vs. Displacement
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: Distance: How far an object has traveled
A scalar quantity: has magnitude only. Displacement: Change in position A vector quantity: has magnitude and direction.

Displacement most common notation or or or

Motion can be described using words.
Motion can be described using diagrams. Motion can be described using equations. Motion can be described using graphs. …but always depends on a FRAME OF REFERENCE. …………………..assign a coordinate system.

Sample 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, Dx = xf – xi Dx = 5.0m – 0.0m Dx = 5.0m

Sample problem #2 Find the displacement of the gecko.

Sample problem #3 Find the displacement of 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 #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, then Dx = 5.0m – 6.0m = -1.0m or 1.0m south of the desk

Sample 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. Displacement
5.0m North

Distance vs. Displacement
6.0m South

Distance vs. Displacement
1.0m South 5.0m North 6.0m South The displacement tells you where the teacher ended up. The distance tells you the total length of his journey.

Changing position over time

How fast is an object moving?
Average Velocity: v displacement : time ratio A vector quantity: has magnitude and direction. total displacement : total time elapsed Average Speed: distance : time ratio A scalar quantity: has magnitude and direction. total distance : total time elapsed

Average Velocity

Velocity vs. Speed Velocity is a vector Velocity = displacement time
Speed is a scalar (direction does not matter) Speed = distance

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

Instantaneous Velocity
Constant Velocity Constant velocity is when an objects velocity remains the same for a given amount of time. Instantaneous Velocity Instantaneous velocity is the velocity at a given point in time. Example: Speedometer, Radar gun

Sample Problem #5 Suzy Physics Student lives 5.0miles south of school.
If she takes 2.0 hours to get to school, what is Suzy’s average velocity?

Sample Problem #6 Suzy Physics Student walks 5.0miles South to school.
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 miles North to IHOP. What is Suzy’s average velocity for the whole trip? What is Suzy’s average speed for the whole trip?

Sample Problem #7

Practice problems

Practice problems

Practice problems

Practice problems

Practice problems

Linear Relationships: y = k x
Slope m=(40-8)/(50-10) m=32/40 m=0.8 g/cm3 Interpolation vs. Extrapolation

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

Graphing Motion Graphs of Position vs. Time Calculate the displacement
Calculate the velocity Describe forward and reverse motion Describe an object staying still

Calculating displacement
A position vs. time graph lets you calculate the displacement between any two moments: x, Position, (m) t, Time, (s)

Calculating displacement
Find the displacement between A and B x, Position, (m) B (1,5) Dx = xf – xi Dx = 5m – 0m Dx = 5m Use Dx = xf –xi Where xf = 5m and xi = 0m t, Time, (s) A (0,0)

Calculating displacement
Find the displacement between C and D x, Position, (m) C (4,8) Use Dx = xf –xi Where xf = 3m and xi = 8m Dx = xf – xi Dx = 3m – 8m Dx = -5m D (8,3) t, Time, (s)

Calculating displacement
Find the displacement from when the object was moving for 2s to when it had been moving for 9s. x, Position, (m) 6m Dx = xf – xi Dx = 6m – 5m Dx = 1m Use Dx = xf –xi Where xf = 6m and xi = 5m 5m t, Time, (s) 2s 9s

Position vs. Time Graph

What are the velocities?

What are the velocities?

Average Velocity on an x-t graph

vav vs vinst on an x-t graph

x-t graph vs motion of a particle

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

Acceleration vs. Acceleration
Terms Acceleration vs. Acceleration Acceleration: The rate at which velocity changes (vector) Acceleration: The rate at which speed changes (scalar) Symbol: a or a SI Unit: meters per second per second or meters per second squared, m/s/s or m/s2

Average Acceleration

Velocity vs. Time Graph The slope of a v vs. t graph is the acceleration The area between the curve and the horizontal axis of a v vs. t graph is the displacement

Instantaneous Acceleration vs Average Acceleration from a v-t graph

v-t graph vs motion of a particle

x-t graph vs motion of a particle with acceleration

What are the accelerations and displacements?

What are the accelerations and displacements?

Acceleration vs. Time Graph
The slope means NOTHING The area between the curve and the horizontal axis is the change in velocity a 10 s t -10 m/s/s

Important Acceleration tells us how fast velocity changes
Velocity tells us how fast position changes

Kinematics Equations (accelerated motion)

Falling Bodies, thrown up objects, and the y-direction
All things fall at the same rate (neglecting air resistance) On earth that rate is 9.80 m/s2 That rate is an acceleration The name of that acceleration is Gravity

Object moving in y-dir