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
Solve for time.
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Multiple both sides by 2
2d = gt2
Divide both sides by g
2d/g = t2
Square root both sides.
T = √(2d/g)
2
t
Example: E = mgh + ½
Solve for m
• E = m(gh + ½ v2)
• m = E/(gh + ½ v2)
2
mv
Example 2: E = mgh + ½
Solve for h
• E – ½ mv2 = mgh
• h = (E – ½ mv2 )/mg
2
mv
Example 3: E = mgh + ½
Solve for v
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E – mgh = ½ mv2
2(E – mgh ) = mv2
2(E – mgh )/m = v2
v = √[2(E – mgh )/m]
2
mv
Example 4: Solve by Substitution
2x + 8y = 1
x = 2y
Unsolvable on its own….
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
x = 2y
y = 1/12
y = 1/6
y = 1/12
x = 2(1/12)
Describing Motion
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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:
a
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
d

d1
 d0
most common
notation
or
or
or
x

xf
 xi
y

y1
 y0
r1
 r0
r

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Motion can be described using words.
Motion can be described using diagrams.
Motion can be described using equations.
Motion can be described using graphs.
http://acme.highpoint.edu/~atitus/physlets/index.html
…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,
x = xf – xi
x = 5.0m – 0.0m
x = 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
x = 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
http://acme.highpoint.edu/~atitus/physlets/index.html
How fast is an object moving?
Average Velocity:
v
v
– displacement : time ratio
– A vector quantity: has magnitude and direction.
– total displacement : total time elapsed
Average Speed:
s
– distance : time ratio
– A scalar quantity: has magnitude and direction.
– total distance : total time elapsed
Average Velocity
v

d
t

d1

d0
t1

t0
Velocity vs. Speed
Velocity is a vector
Velocity = displacement
time
Speed is a scalar (direction does not matter)
Speed = distance
time
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?
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
50
m=(40-8)/(50-10)
Mass(g)
40
m=32/40
30
m=0.8 g/cm3
20
10
0
10
20
30
Volume(mL)
40
Interpolation
vs.
Extrapolation
Graphing Motion
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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
1. Calculate the displacement
2. Calculate the velocity
3. Describe forward and reverse motion
4. Describe an object staying still
Calculating displacement
x, Position, (m)
A position vs. time graph lets you calculate the
displacement between any two moments:
t, Time, (s)
Calculating displacement
x, Position, (m)
Find the displacement between A and B
A (0,0)
B (1,5)
Use x = xf ––xxii
Where
5m– 0m
xx=
f =5m
and xx=i =5m
0m
t, Time, (s)
Calculating displacement
Find the displacement between C and D
x, Position, (m)
C (4,8)
Use x
x == xxff –x
– xi i
Wherexxf==3m
3m– 8m
and xxi==-5m
8m
t, Time, (s)
D (8,3)
Calculating displacement
x, Position, (m)
Find the displacement from when the object was
moving for 2s to when it had been moving for 9s.
6m
Use x = xf ––xxii
5m
Where
6m– 5m
xx=
f =6m
and xx=i =1m
5m
2s
t, Time, (s)
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
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Motion can be described using words.
Motion can be described using diagrams.
Motion can be described using equations.
Motion can be described using graphs.
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
-10 m/s/s
t
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