Welcome to AP Physics! Mr. Clark will be holding a Math Summer Bridge Camp this summer
to help students learn some math to help lessen the beat down you will experience in the
course. Mr. Clark and I highly recommend you attend this 2 hour session during the June
intersession. The class will pay dividends in the long run.
1. The AP exams are in early May which requires a very intense pace. We will start on the
physics subject matter the very first day of school. This assignment will help with valuable
skills which you will use throughout the AP course.
2. AP Physics requires a proficiency in algebra, trigonometry, and geometry. AP physics is a
course in applied mathematics. The assignment includes mathematical problems which
are considered routine in AP Physics. This includes key metric system conversion factors
and how to use them, understanding vectors, and graphing.
3. The math review is to be completed with all work shown. No miracles!!
a. The following are ordinary physics problems. Place the answer in scientific notation when
appropriate; for example, 200 is easier to write than 2.00 x 10 2, but 2.00 x 108 is easier to
write than 200,000,000. Do your best to cancel units and show the simplified units in the
final answer. Do the math and record the answer here.
a. 𝑇𝑠 = 2 ∙ 𝜋 ∙ √
0.045 𝑘𝑔
𝑘𝑔
2000 ⁄ 2
𝑠
=
b. 𝐾 = 0.5 ∙ 660 𝑘𝑔 ∙ (2.11 𝑥 104 𝑚⁄𝑠)2 =
c. 𝐹 =
d.
1
𝑅𝑝
=
e. 𝑒 =
f.
9𝑥 109
1
450 Ω
𝑁∙𝑚2
∙ 3.2 𝑥10−9 𝐶 ∙ 9.6 𝑥 10−9 𝐶
𝐶2
(0.32 𝑚)2
+
1
=
940 Ω
1700 𝐽−330 𝐽
1700 𝐽
=
=
1.33 ∙ sin 25 = 1.50 ∙ sin 𝜃 ; find θ
g. 𝐾𝑚𝑎𝑥 = (6.63 𝑥 10−34 𝐽 ∙ 𝑠 ∙ 7.09 𝑥 1014 𝑠) − 2.17 𝑥 10−19 𝐽
h. 𝛾 =
1
2.25 𝑥 108 𝑚⁄𝑠
√1−
3 𝑥 108 𝑚⁄𝑠
b. Problems on the AP exam are often done with variables only. Solve for the variable indicated.
Don’t let the different letters confuse you. Manipulate the equations algebraically as if the
variables were numbers. This one will have to be done on a separate sheet of paper.
a. 𝑣 2 = 𝑣𝑜2 + 2 ∙ 𝑎 ∙ (𝑠 − 𝑠𝑜 ) for a:
1
b. 𝐾 = ∙ 𝑘 ∙ 𝑥 2 for x:
2
𝑙
c. 𝑇𝑝 = 2 ∙ 𝜋 ∙ √ for g:
𝑔
d. 𝐹𝑔 =
𝐺∙𝑚1 ∙𝑚2
𝑟2
for r:
1
e. 𝑚 ∙ 𝑔 ∙ ℎ =
f.
1
2
∙ 𝑚 ∙ 𝑣 2 for v:
1
𝑥 = 𝑥𝑜 + 𝑣𝑜 ∙ 𝑡 + ∙ 𝑎 ∙ 𝑡 2 for t:
g. 𝐵 =
𝜇𝑜 ∙𝐼
2
for r:
2∙𝜋∙𝑟
𝑚∙𝜆∙𝐿
h. 𝑥𝑚 =
for d:
𝑑
i. 𝑃 ∙ 𝑉 = 𝑛 ∙ 𝑅 ∙ 𝑇 for T:
𝑛
j. sin 𝜃𝑐 = 1 for θc
k. 𝑞 ∙ 𝑉 =
l.
1
𝑓
=
1
𝑑𝑜
1
2
+
𝑛2
∙ 𝑚 ∙ 𝑣 2 for v:
1
𝑑𝑖
for di
c. Science uses the KMS or SI units of measure. KMS stands for kilogram, meter, second. The
equations in physics depend on unit agreement, so you must convert to KMS in most
problems to arrive at the correct answer. Other conversions will be presented as needed.
Examples of common conversions:
a. Kilometers (km) to meters (m) and meters to kilometers
b. Centimeters (cm) to meters (m) and meters to centimeters
c. Millimeters (mm) to meters (m) and meters to millimeters
d. Micrometers (μm) to meters (m) and meters to micrometers
e. Nanometers (nm) to meters (m) and meters to nanometers
f. Gram (g) to kilogram (kg)
g. Celsius (°C) to Kelvin (K)
h. Atmosphere (atm) to Pascals (Pa)
i. Liters (L) to cubic meters (m3)
If you don’t know the conversion factors, they are available online. Colleges and employers
need students to be able to find their own information – so here is some practice! This one will
have to be done on a separate sheet of paper.
a. 4008 g to kg
b. 1.2 km to m
c. 823 nm to m
d. 298 K to °C
e. 0.77 m to cm
f. 8.8 x 10-8 m to mm
g. 1.2 atm to Pa
h. 25 μm to m
i. 2.65 mm to m
j. 8.23 m to km
k. 5.4 L to m3
l. 40.0 cm to m
m. 6.23 x 10-7 m to nm
n. 1.5 x 1011 m to km
d. Solve the geometry problems. You can do these on this page.
a. Line B touches the circle at a single point. Line
A extends through the center of the circle.
i. What is line B called in reference to the circle?
ii. How large is the angle between lines A and B?
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b. What is angle C?
c. How large is angle θ?
d. The radius of a circle is 5.5 cm.
i. What is the circumference in meters?
ii. What is its area in m2?
e. What is the area under the curve
at the right?
Calculate the area of the following shapes. It may be necessary to break up the figure into
common shapes.
3
Calculate the unknown angle values for questions 3-6
Right Triangles & Trigonometry
Suppose you wanted to get from point A to point C in the
following diagram. You could go directly from A to C, or
you could go to the right from A to B and then go
straight up from B to C. Thus the direction we go is
important. We will define the distance from C to A as
our displacement. You’ll see shortly that the
displacement is a vector. In many cases, we will be
interested in the x and y component of a vector. In this
case, the x component is AB. The y component is BC.
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Pythagorean Theorem
If two of the three sides of a right triangle
are known, we can find the 3rd side using
the Pythagorean Theorem.
Recall that a2 + b2 = c2
122 + 52 = c2
144 + 25 = c2
169 = c2
13 = c
Right Triangle Trigonometry
Both of the triangles shown are right triangles. Angle α is the same for both. Side
c is the hypotenuse. Side a is adjacent to angle α. Side b is opposite angle a. If we
divided side b by side a for both triangles we would get the same number. The only
way we could get a different ratio would be if the angle changed. For example, if
the angle increased, side b would have to increase, while side a remained the same.
That would cause the ratio to increase. We call the ratio of side b to side a the
tangent of the angle. The same logic is true for the ratios of any two sides. We will
use three ratios:
An easy way to remember these is the acronym SOH CAH TOA (pronounced so-ca-toe-a):
SOH → Sine is Opposite / Hypotenuse
CAH → Cosine is Adjacent / Hypotenuse
TOA → Tangent is Opposite / Adjacent
Make sure the calculator is in the degree mode.
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Example: find sides b and c.
tan 𝛼 =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑏
=
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
𝑎
tan 25 =
𝑏
10
𝑏 = 10 ∙ tan 25 = 4.66
Now that you know side b, you can use trig or the Pythagorean theorem to find side c.
Finding angles
Since each angle has a unique sine, cosine and
tangent value, we can use these values to find the
angle. We call these functions the inverse tangent,
inverse sine, or inverse cosine.
𝛼 = 𝑡𝑎𝑛−1
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑏
= 𝑡𝑎𝑛−1
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
𝑎
This is read as: α is the angle whose tangent is b/a.
Example: 𝛼 = 𝑡𝑎𝑛−1
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
= 𝑡𝑎𝑛−1
5
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= 22.62°; the inverse functions are normally above the
standard trig function on the calculator and you have to use the shift key or 2nd key.
For the right triangle to the right, determine:
a. the length of side a?
b. the sine of angle α?
c. the cosine of angle a?
d. angle α = ______________degrees.
e. angle β = ______________degrees.
(use trig and check to see if all the angles add to 180o)
Calculate the following unknowns using trigonometry. Use a calculator, but show all of your
work. Please include appropriate units with all answers. (Watch the unit prefixes!)
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You will need to be familiar with trigonometric values for a few common angles. Memorizing
this unit circle diagram in degrees or the chart below will be very beneficial for next year in
both physics and pre-calculus. How the diagram works is the cosine of the angle is the xcoordinate and the sine of the angle is the y-coordinate for the ordered pair. Write the ordered
pair (in fraction form) for each of the angles shown in the table below
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Refer to your completed chart to answer the following questions.
10. At what angle is sine at a maximum?
a. At what angle is sine at a minimum?
b. At what angle is cosine at a minimum?
c. At what angle is cosine at a maximum?
d. At what angle are the sine and cosine equivalent?
e. As the angle increases in the first quadrant, what happens to the cosine of the angle?
f. As the angle increases in the first quadrant, what happens to the sine of the angle?
Use the figure to the right to answer the next two questions.
a. Develop an equation for h in terms of l and θ.
b. What is the value of h if l = 6 m and θ = 40°.
Vectors
Two videos which may help you with working with vectors.
http://www.khanacademy.org/science/physics/v/introduction-to-vectors-and-scalars
http://www.khanacademy.org/science/physics/v/visualizing-vectors-in-2-dimensions
Most of the quantities in physics are vectors. This makes proficiency in vectors extremely
important.
Magnitude: Size or extent. The numerical value only.
Direction: Alignment or orientation of any position with respect to any other position.
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Scalars: A physical quantity described by a single number and units. A quantity described by
magnitude only; a number and unit only. Examples: time, mass, and temperature.
Vector: A physical quantity with both a magnitude and a direction. A directional quantity.
Examples: velocity, acceleration, force.
Notation: → or
𝐴
→
𝐴
Length of the arrow is proportional to the vectors magnitude. Direction the arrow points is the
direction of the vector.
Negative Vectors
Negative vectors have the same magnitude as their positive counterpart. They are just pointing
in the opposite direction.
–→
𝐴
Vector Addition and subtraction
Think of it as vector addition only. The result of adding vectors is called the resultant →.
𝑅
So if A has a magnitude of 3 and B has a magnitude of 2, then R has a magnitude of 3 + 2 = 5.
When you need to subtract one vector from another think of the one being subtracted as being
a negative vector. Then add them.
A negative vector has the same length as its positive counterpart, but its direction is reversed.
So if A has a magnitude of 3 and B has a magnitude of 2, then R has a magnitude of
3 + -2 = 1.
This is very important. In physics a negative number does not always mean a smaller
number. Mathematically –2 is smaller than +2, but in physics these numbers have the same
magnitude (size), they just point in different directions (180° apart).
There are two methods of adding vectors:
1. Parallelogram:
A+B
A–B
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2. Tip to Tail
A+B
A–B
It is readily apparent that both methods arrive at the exact same solution since either method
is essentially a parallelogram. It is useful to understand both systems. In some problems one
method is advantageous, while in other problems the alternative method is superior.
e. Draw the resultant vector using the parallelogram method of vector addition. You can answer
these on this page.
Example:
a.
c.
b.
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d.
e.
Direction: What does positive or negative direction mean? How is it referenced? The answer
is the coordinate axis system. In physics a coordinate axis system is used to give a
problem a frame of reference. Positive direction is a vector moving in the positive x or
positive y direction, while a negative vector moves in the negative x or negative y direction.
This also applies to the z direction, which will be used sparingly in this course.
What about vectors that don’t fall on the axis? You must specify their direction using degrees
measured from one of the four axes (-x, +x, -y, +y; or E, N, S, and W).
Component Vectors
A resultant vector is a vector resulting from the sum of two or more other vectors.
Mathematically the resultant has the same magnitude and direction as the total of the vectors
that compose the resultant. Could a vector be described by two or more other vectors? Would
they have the same total result? This is the reverse of finding the resultant. You are given the
resultant and must find the component vectors on the coordinate axis that describe the
resultant.
Any vector can be described by an x axis vector and a y axis vector which summed together
mean the exact same thing. The advantage is you can then use plus and minus signs for
direction instead of the angle.
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f.
For the following vectors draw the component vectors along the x and y axis. You can answer
on this page.
Obviously the quadrant that a vector is in determines the sign of the x and y component
vectors.
How are vectors used in Physics? They are used everywhere!
Speed: Speed is a scalar. It only has magnitude (numerical value). Vs = 10 m/s means that
an object is going 10 meters every second. But, we do not know where it is going.
Velocity: Velocity is a vector. It is composed of both magnitude and direction. Speed is a part
(numerical value) of velocity. v = 10 m/s north, or v = 10 m/s in the +x direction, etc.
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There are three types of speed and three types of velocity.
1. Instantaneous speed / velocity: The speed or velocity at an instant in time. You look
down at your speedometer and it says 20 m/s. You are traveling at 20 m/s at that instant.
Your speed or velocity could be changing, but at that moment it is 20 m/s.
2. Average speed / velocity: If you take a trip you might go slow part of the way and fast at
other times. If you take the total distance traveled divided by the time traveled you get the
average speed over the whole trip. If you looked at your speedometer from time to time you
would have recorded a variety of instantaneous speeds. You could go 0 m/s in a gas
station, or at a light. You could go 30 m/s on the highway, and only go 10 m/s on surface
streets. But, while there are many instantaneous speeds there is only one average speed
for the whole trip.
3. Constant speed / velocity: If you have cruise control you might travel the whole time at
one constant speed. If this is the case then you average speed will equal this constant
speed.
A trick question: Will an object traveling at a constant speed of 10 m/s also always have
constant velocity?
Not always. If the object is turning around a curve or moving in a circle it can have a constant
speed of 10 m/s, but since it is turning, its direction is changing. And if direction is changing
then velocity must change, since velocity is made up of speed and direction.
Constant velocity must have both constant magnitude and constant direction.
Rate: Speed and velocity are rates. A rate is a way to quantify anything that takes place
during a time interval. Rates are easily recognized. They always have time in the denominator.
The very first Physics Equation: Velocity and Speed both share the same equation.
Remember speed is the numerical (magnitude) part of velocity. Velocity only differs from speed
in that it specifies a direction.
𝑣=
𝑥
𝑡
; v stands for velocity; x stands for displacement; t stands for time
Displacement is a vector for distance traveled in a straight line. It goes with velocity. Distance
is a scalar and goes with speed. Displacement is measured from the origin. It is a value of
how far away from the origin you are at the end of the problem. The direction of a
displacement is the shortest straight line from the location at the beginning of the problem to
the location at the end of the problem.
How do distance and displacement differ? Supposes you walk 20 meters down the + x axis
and turn around and walk 10 meters down the – x axis. The distance traveled does not depend
on direction since it is a scalar, so you walked 20 + 10 = 30 meter. Displacement only cares
about you distance from the origin at the end of the problem. +20 – 10 = 10 meter.
g. Attempt to solve the following problems. Answer these on a separate sheet of paper. Always
use the KMS system: Units must be in kilograms, meters, seconds.
On the all tests, including the AP exam you must:
1. List the original equation used.
2. Show correct substitution.
3. Arrive at the correct answer with correct units.
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a. A car travels 35 km west and 75 km east. What distance did it travel?
b. A car travels 35 km west and 75 km east. What is its displacement?
c. A car travels 35 km west, 90 km north. What distance did it travel?
d. A car travels 35 km west, 90 km north. What is its displacement?
e. A bicyclist pedals at 10 m/s in 20 s. What distance was traveled?
f. An airplane flies 250.0 km at 300 m/s. How long does this take?
g. A skydiver falls 3 km in 15 s. How fast are they going?
h. A car travels 35 km west, 90 km north in two hours. What is its average speed?
i.
A car travels 35 km west, 90 km north in two hours. What is its average velocity?
Graphing and Functions
A greater emphasis has been placed on conceptual questions and graphing on the AP exam.
Below you will find a few example concept questions that review foundational knowledge of
graphs. Ideally you won’t need to review, but you may need to review some math to complete
these tasks. At the end of this part is a section covering graphical analysis that you probably
have not seen before: linear transformation. This analysis involves converting any non-linear
graph into a linear graph by adjusting the axes plotted. We want a linear graph because we
can easily find the slope of the line of best fit of the graph to help justify a mathematical model
or equation.
Key Graphing Skills to remember:
1. Always label your axes with appropriate units.
2. Sketching a graph calls for an estimated line or curve while plotting a graph requires
individual data points AND a line or curve of best fit.
3. Provide a clear legend if multiple data sets are used to make your graph understandable.
4. Never include the origin as a data point unless it is provided as a data point.
5. Never connect the data points individually, but draw a single smooth line or curve of best
fit.
6. When calculating the slope of the best fit line you must use points from your line. You may
only use given data points IF your line of best fit goes directly through them.
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Conceptual Review of Graphs
Two graphs are shown. Is the slope of
the graph
i.
i. greater in Case A,
ii.
ii. greater in Case B,
iii. the same in both cases?
Four points are located on
a graph.
Rank the slopes of the
graph at the labeled points
below.
Explain your reasoning.
Line Data Graph Interpretation
15
A student makes the following claim about some data
that he and his lab partners have collected.
“Our data show that the value of y decreases as x
increases. We found that y is inversely proportional
to x.”
What, if anything, is wrong with this statement? If
something is wrong, identify and explain how to
correct all errors. If this statement is correct, explain
why.
Linear and Non-Linear Functions
You must understand functions to be able to linearize. First let’s review what graphs of certain
functions look like. Sketch the shape of each type of y vs. x function below. The letter k is
listed as a generic constant of proportionality.
You will notice that only the linear function is a straight line. We can easily find the slope of
our line by measuring the rise and dividing it by the run of the graph or calculating it using
two points. The value of the slope should equal the constant k from the equation. Finding k is
a bit more challenging in the last three graphs because the slope isn’t constant. This should
make sense since your graphs aren’t linear. So how do we calculate our constant, k? We need
to transform the nonlinear graph into a linear graph in order to calculate a constant slope. We
can accomplish this by transforming one or both of the axes for the graph. The hardest part is
figuring out which axes to change and how to change them. The easiest way to accomplish this
task is to solve your equation for the constant k. Note in the examples from the last page there
is only one constant, but this process could be done for other equations with multiple
constants. Instead of solving for a single constant, put all of the constants on one side of the
equation. Let’s try a few problems. Solve each of the equations for the constant(s).
1. Σ𝐹 = 𝑚 ∙ 𝑎
2. 𝐹 =
3. 𝑈𝑠 =
𝐺∙𝑀∙𝑚
𝑟2
1
2
(m and a are variables)
(F and r are variables)
∙ 𝑘 ∙ 𝑥2
(Us and x are variables)
16
When you solve for the constant, the other side of the equation should be in fraction form.
This fraction gives the rise and run of the linear graph. Whatever is in the numerator is the
vertical axis and the denominator is the horizontal axis. If the equation is not in fraction form,
you will need to inverse one or more of the variables to make a fraction. Two of the above
equations will need to be inverted to get into fraction form. Invert each of the above equations
as necessary to get a fraction. The expression opposite the fraction is equal to the slope of the
graph. Since the slope equals constants, a linear graph will always be produced. First let’s
solve each equation to figure out what we should graph. Then look below at the example and
complete the last one, a sample AP question, on your own.
State what should be graphed in order to produce a linear graph to solve for k.
Chemistry Example: Let’s look at an equation you should remember from chemistry.
According to Boyle’s law, an ideal gas obeys the following equation 𝑃1 ∙ 𝑉1 = 𝑃2 ∙ 𝑉2 = 𝑘 . This
states that pressure and volume are inversely related, and the graph on the left shows an
inverse shape. Although the equation is equal to a constant, the variables are not in fraction
form. One of the variables, pressure in this case, is inverted. This means every pressure data
point is divided into one to get the inverse. The graph on the left shows the linear relationship
between volume V and the inverse of pressure 1/P. We could now calculate the slope of this
linear graph.
17
a) Draw a line of best fit for the distance vs. time graph above.
b) If only the variables D and t are used, what quantities should the student graph in order to
produce a linear relationship between the two quantities?
c) On the grid below, plot the data points for the quantities you have identified in part (b), and
sketch the straight-line fit to the points. Label your axes and show the scale that you have
chosen for the graph.
d) Calculate the value of g by using the slope of the graph.
Relationships from equations
26.) Consider
z
y
x
and
a  bc,
a.) As x increases and y stays constant, z _____________________.
b.)


As y increases and x stays constant, z _____________________.
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c.) As x increases and z stays constant, y _____________________.
d.) As a increases and c stays constant, b _____________________.
e.) As c increases and b stays constant, a _____________________.
f.) As b increases and a stays constant, c _____________________
Systems of equations
Use the equations in each problem to solve for the specified variable in the given terms.
Simplify.
27.) Ff    FN and FN  m  g  cos  . Solve for  in terms of Ff, m, g, and .
28.)
F1 + F2 = FT
and
F1 × d1 = F2 × d2.
Solve for F1 in terms of FT, d1, and d2.
Graphing Equations
29.) On the y vs. x graphs below, sketch the relationships given.
a.)
y = mx + b, if m > 0 and b = 0.
b.)
y = x2
c.)
y= x
d.)
y = 1/x
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e.)
y = 1/x2
f.)
y=
1
x
20
Uncertainty
1. What is meant by “uncertainty” in a measurement? Why do you think it is important to
include an indication of uncertainty when you report experimental results?
2. What is systematic (or systemic) error in an experiment? In what ways would it affect the
data? Give one example.
3. How are analog measuring devices different from digital devices?
4. What is random error in an experiment? In what ways would it affect the data? Give one
example.
5. How is uncertainty represented on a graph?
6. Suppose you are measuring the length and the width of a piece of paper with a metric ruler,
on which the smallest division is the millimeter (0.001 m). You measure the length to be
0.0130 m and the length to be 0.085 m.
a. What would be the appropriate way to report each measurement, including the
uncertainty?
b. What would be the area of the piece of paper, with the correct uncertainty value
included in your answer?
7. Briefly describe an acceptable method for determining the uncertainty when using multiple
repeated measurements.
8. In your own words, compare and contrast precision and accuracy.
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