AP Physics Math Review Part 1
pdf
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School
Laramie County Community College *
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Course
2010 500
Subject
Mathematics
Date
Nov 24, 2024
Type
Pages
13
Uploaded by computerwizard697
Name_____________________
Hr_____
Date_________
►Algebra: Solve for the Numeric Answer.
Equations are used in physics to represent relationships. Students have a good understanding
of how to solve equations by plugging in numeric values for variable.
PRACTICE:
Using algebra, solve for the indicated variable. Be sure to show all steps required
to solve for
variable, including substitution. Please include the units as they work out
algebraically.
1. K = ½ mv
2
m = 210 kg, v = 10.5 m/s
Answer: _________ Units: kgm^2/s^2
2. F =
𝐺𝑀
1
𝑀
2
?
2
G = 6.67 x 10
-11
Nm
2
M
1
= 5.64 x 10
24
kg
M
2
= 1.99 x 10
31
kg
r = 1.51 x 10
15
m
Answer: _________ Units: ________
3.
1
𝑅
𝑒?
=
1
𝑅
1
+
1
𝑅
2
R
1
= 24 Ω; R
2
= 18 Ω
Answer: _________ Units: ________
4.
τ
= r Fsin θ
r = 1.4 m; F = 28 N; θ = 47°
Answer: _________ Units: ______
(HINT: check calculator mode.)
5. T = 2
π
?
𝑔
l
=
0.34 m; g = 9.8 m/s
2
Answer: _________ Units: ________
►Algebra: Solve for the Variable.
In physics, we are less interested in the numeric solution, and more interested in the
relationship between variables. Solving for variables will be a daily routine in this class. Don’t
get confused with the letters, think of them as numbers and algebraically rearrange for the
chosen variable
1. U
g
= mgh; solve for h
2. F = kq
1
q
2
; solve for q
2
r
2
3. P =
W ; solve for ∆ t
4. R = ρ
; solve for ρ
?
𝑎
∆ t
5. a
c
= v
2
; solve for v
6. v
f
2
= v
0
2
+ 2a(x
f
– x
0
) ;solve for x
0
r
7. qV = ½ mv
2
; solve for v
8.
T = 2
π
; solve for k
?
?
Use these equations in combination to solve for the following variable:
9. L = I ω
and
L =
τ
t
solve for torque
τ
in terms of I, ω, and t
10. K = ½ I ω
2
v = ω r and
I = ½ mr
2
solve for K in terms of m, v and any constants.
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Equations of Linear Relationships
A linear relationship
(or linear
association) is a statistical term used to describe a
straight-line relationship
between a variable and a constant. Linear relationships
can be
expressed either in a graphical format or as a mathematical equation of the form y = mx + b.
●
Therefore, the variable on the left-hand side, y, belongs on the y axis.
●
The variable on the right-hand side, x belongs on the x axis.
●
The variable m is slope and is a constant value coefficient. The slope will have the units
of the y axis as the numerator and the units of the x axis as the denominator.
●
The variable b is the y-intercept. It indicates the initial, or starting, value for the y
variable. The y-intercept b will also have the same units as the y variable.
Graphs of Linear Relationships
Graph shape
Written relationship
Modification required
to linearize graph
General Algebraic
representation
As x increases, y increases
proportionally.
Y is directly proportional to
x.
None
Look at the example below. The variable on the x axis is time, the variable on the y axis is
distance, slope is the time rate of change of distance or distance/time and has units of mi/hr,
and the y intercept is 0 miles, indicating that the starting position was at the origin.
Relationship statement:
The graph of distance vs. time shows a linear relationship. As time
increases, distance increases proportionally. Distance is directly proportional to time.
General Equation:
y = mx + b
Specific Equation:
d = (25mi/hr)t + 0 mi
(The 0 mi is not necessary here but added for clarity to show the y-intercept with units. Here I
have included the variable and the units of the variables.)
Extended Question #1:
If time t = 8 hours, what is the distance traveled?
d = (25mi/hr) t
d = (25 mi/hr) (8hr)
d = 200 mi
Extended Question #2:
If the distance traveled is 400 miles, how much time elapsed?
d = (25mi/hr) t
400 mi = (25mi/hr) t
400 mi/(25 mi/hr) = t
t = 16 hours
PRACTICE:
Circle or highlight the equations that represent a linear relationship with the
variable on the left-hand side and time (t).
x = (v
o
+ v
f
)t
2
PRACTICE:
Suppose you recorded the following data during a study of the relationship of force
and acceleration. Prepare a graph showing these data on a separate sheet of paper (See
#12). Attach to this worksheet for grading.
*On all activities within this course, the independent variable is always listed first* 1. What is the variable on the x axis? ____________________________________________
2. What are the units of the x axis? ______________________________________________
3. What is the variable on the y axis? ____________________________________________
4. What are the units of the y axis? ______________________________________________
5. Describe the relationship between force and acceleration as shown by the graph. ______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
____________________________________
6. What is the slope of the graph? Remember to include the units with your slope. One newton
(N) equals 1
kg*m/s
2
. Slope =
7. What physical quantity does the slope represent? Use algebra to analyze the units to find
out.
8. What is the general equation for the line? 9. Write the specific equation for the line. 10. Using the equation you generated in #9 above, what is the value of the force for an
acceleration of 15 m/s
2
? 11. Using the equation you generated in #9 above, what is the acceleration when the force is
50.0 N?
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12. Graph the data. We will do it by hand as well as on the computer in this class.
Points to remember:
●
Be sure you have included a title in the form of DV vs. IV.
●
Make sure to label the x axis with units.
●
Make sure to label the y axis with units. ●
Make sure the graph is properly scaled.
●
Make sure to have a best fit line. ●
Make sure your graph takes up the majority of the given space**
Equations of Quadratic (Square) Relationships
A quadratic relationship
is a mathematical relation
between two variables that follows the
form of a quadratic
equation.
Quadratic relationships
can be expressed either in a graphical
format or as a mathematical equation. To put it simply, the equation that holds our two variables
looks like the following: Here, y and x are our variables, and a, b, and c are constants. y = ax
2
+
bx + c.
●
Therefore, the variable on the left-hand side, y, belongs on the y axis.
●
The variable on the right hand side, x belongs on the x axis.
●
The variable c is the y-intercept. It indicates the initial, or starting, value for the y
variable. The y-intercept c will also have the same units as the y variable.
Graphs of Quadratic Relationships
Graph shape
Written relationship
Modification required
to linearize graph
General Algebraic
representation
Y is proportional to the
square of x.
Graph y
vs x
2
Look at the example below. The variable on the x axis is time, the variable on the y axis is
displacement. How can I calculate the slope of this line? Newton would say to take the
derivative. But in AP Physics 1, we have to use algebra. Instead, I look at the form of the graph
and identify it as a quadratic. I then square the values of the x axis, in this case, square time,
and re-graph.
Linearized Graph:
Relationship statement:
The graph of distance vs. time shows a quadratic relationship. As
time Increases, distance increases exponentially. Distance is proportional to the square of time..
General Equation:
y = mx
2
+ b
Specific Equation:
d = (2.45 m/s
2
)t
2
+ 2 m
(The y-intercept shows that the motion did not start at the origin but started at the 2 m
mark).
Extended Question #1:
If time t = 60s, what is the distance traveled?
d = (2.45 m/s
2
)t
2
+ 2 m
d = (2.45 m/s
2
)(60s)
2
+ 2 m
d = 8,822 m
Extended Question #2:
If the distance traveled is 400 m, how much time elapsed?
d = (2.45 m/s
2
)t
2
+ 2 m
400 m = (2.45 m/s
2
)t
2
+ 2 m
(400m – 2m) = (2.45 m/s
2
)t
2
388 m/(2.45 m/s
2
) = t
2
950.6 s
2
= t
2
t = 30.8 s
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PRACTICE:
Circle or highlight the equations that represent a Square relationship with the
variable on the left-hand side.
x = (v
o
+ v
f
)t
2
PRACTICE:
Suppose you recorded the following data during a study of the relationship
of distance and time.The following data show the distance an object travels in certain
time periods.
Prepare
a
graph
showing
these
data
on
a
separate
sheet
of
paper
(See
#12). Attach
to
this worksheet for grading.
1. What is the variable on the x-axis?
2. What are the units of the x-axis? 3. What is the variable on the y-axis? 4. What are the units of the y-axis?
5. Describe the relationship between x and y as shown by the graph. 6. What is the general equation for the graph? 7. Is the change in distance traveled greater between 0 s and 1 s or 3 s and 4 s?
8. Describe the steps required to linearize the graph.
9. Graph the original data. .
Then, linearize the data
For both graphs, ●
Be sure you have included a title in the form of DV vs. IV.
●
Make sure to label the x axis with units.
●
Make sure to label the y axis with units.
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10. Check the variable that changed. Are the units and variable adjusted to show the
new relationship? What are the units of slope, units of x-axis, and units of y-axis?
Relationship:
Units of Slope:
Units of Linearized x-axis:
Units of Linearized y-axis:
(This does not imply that both axes required a change.)
11. What is the specific equation for the linearized graph (don’t forget units of slope and
any y-intercept)? _______________d = 3m/s^2*t^2 + 0 m_____________________________
12. Using the equation you generated in #11 above, what is the value of the distance for
a time of 7 s? (Show work starting with the Equation from #11)
13. Using the equation you generated in #11 above, what is the value of time for a
distance of 80 m?
(Show work starting with the Equation from #11)