Honors Physics
Summer Assignment: Mathematics in Physics
Mr. Wright
To the prospective Honors Physics student,
Welcome to the Honors Physics summer assignment!
As an upper classmen in an honors science class, it is understood that you have an
interest in pursuing a career in some science based pursuit. Whether it be medicine or
healthcare, engineering or research science, the Honors Physics experience is designed to
provide you with a sound foundation in the fundamental models that govern the behavior
of all physical systems.
The summer assignment focuses on reviewing the mathematics you have already
mastered in your studies as underclassmen. If there is any topic presented in this
assignment that baffles you, then during the summer months is the time to review that
material so you can hit the ground running in September.
Also presented in the summer assignment, is an introduction to the language of Physics.
Vector mathematics may sound intimidating at first but you have already been introduced
to the techniques in your study of right triangle geometry. The sine, cosine, and tangent
functions and the Pythagorean Theorem are already familiar to you, in Physics we simply
use different terminology to represent physical quantities that can be manipulated using
vector mathematics.
The problems included in the summer assignment will be collected Sept. 4, 2015.
Do not work alone! Find a physics buddy, doing physics alone can be tedious and
tiresome. Success in Honors Physics is dependent upon proper presentation of concepts
and techniques (which I guarantee) and a willingness on the part of the student to be
eager to learn and work, (which only you can guarantee). The study of physics is an
exercise in intellect; I trust you will find the experience mentally stimulating and above
all…FUN!
Enjoy your summer, see you in September!
Welcome to Honors Physics,
Mr. Wright
Physics instructor
TWP Science
The Mathematics You Know
I.
II.
III.
IV.
Scientific Notation
1. writing numbers using exponential notation
2. manipulating exponential notation during algebraic analysis
Basic rules of algebraic manipulation
1. manipulating equations with variables only
2. recognizing proper order of operations
3. proper use of the quadratic equation
Graphing Analysis
1. plotting coordinates
2. determining slopes
Basic right triangle geometry
1. familiarity with 3-4-5 triangle
2. Pythagorean Theorem
3. SohCahToa (sine, cosine, tangent functions)
Vector Mathematics
At some point early on in their study of Physics most students begin to wonder why this
course is considered a science class as opposed to another math class…the difference is
subtle but important. In science we use mathematics to create accurate, efficient models
of physical systems that possess predictive capability…for instance: how high into the
air can a person throw a ball? Well, some factors that will affect the outcome of the
thrown ball (known as parameters) are things like how fast is it thrown or are there any
physical effects opposing the ball’s upward motion.
Just remember that in Physics, all the letters and numbers represent actual measured
physical quantities, such as velocity, acceleration or force. This is why a number without
appropriate units is meaningless; something to be avoided at all costs!
To mathematically describe the motion of an object we employ the techniques of a
branch of Physics called “kinematics.” We remember that we can pinpoint the location
of an object by either using a Cartesian coordinate system (aka rectangular coordinates)
or polar coordinates. We also know how to convert from one system to the other using
Pythagoras and trigonometry. The twist for Physics is that we use geometry to
represent the actual displacement, velocity, and acceleration of an object experiencing
motion by employing vectors.
A vector looks like a ray but the length of the line segment drawn to scale represents the
magnitude or size of some physical quantity and the arrowhead indicates direction for the
physical quantity that possess both magnitude and direction…such as displacement,
velocity, or acceleration.
Some examples: in which 1cm=1m or 1cm=1m/s, note that a vector has magnitude and
direction and also note that a vector represents a physical quantity or measured value…a
number without units indicating the quantity being described is meaningless!
Displacement vector…representing 3 meters to the left…
-3m
Displacement vector…representing 3 meters to the right……
3m
Velocity vector representing 5m/s to the right……
5 m/s
Now…if you experience a series of changes in position (known as displacement) you can
add up the vectors…2m to the right and 3m to the right and you end up 5m to the right of
your original position…however, 2m to the right and 3m to the left and you end up 1m to
the left of your original position…a displacement of -1 meter indicates a change of
position to the left…there is no such thing as moving less than 0 meters so the negative
sign indicates a direction that is opposite the direction we usually treat as
positive….hmmmm.
All this “to the right” and “to the left” is ridiculously inefficient and what happens when
we want to go up or down or in and out??? Let us use “unit vectors.”
A unit vector has a magnitude of 1…hence the term “unit.” Imagine a traditional
Cartesian coordinate system…the positive “x” direction which points along the axis
known as the abscissa will be represented by a vector of magnitude, 1, and labeled i
So a displacement 3m to the right is now written as 3i meters. In texts, a vector is usually
written in bold print. The “3” indicates the magnitude and the unit vector, i represents the
positive “x” direction. The positive “y” direction usually associated with “up” and
pointing along the axis known as the ordinate is represented by the unit vector, j, and out
of the plane of the page is the k direction. (The axis pointing in the “z” direction is called
the applicate, but don’t panic!, most of the time we will work in only 2 dimensions.)
(Don’t Panic!!!....turn the page!)
Good!
+
=
2i m
3i m
5i m
+
=
-i m
2i m
- 3i m
Easy…Now, what do you get when you add a displacement of 3i m to 4j m?
Just use your knowledge of mathematics!
? m
4j m
5 m at 53 degrees
Or equivalently (3i + 4j ) m
3i m
Alright! You have been introduced to vector mathematics. It really is a piece of pie
because your mathematics teachers have already taught you the mechanics of vector math
they just do not call it vector math! In mathematics we have a hypotenuse and legs of a
right triangle…whereas in Physics we have a resultant and components.
Geometrically, place vectors head to tail when adding them together and use
trigonometry to determine the resultant.
Problems
1. Write the number in proper scientific notation.
(a) 725
(b) 300,000,000
(c) 0.00000950
2. Provide the proper response.
(Note: use a calculator if you must, but you should also be able to complete both
without using a calculator!)
(a) (3 x 107)(6 x 10-12) =
(b) (3 x 106 )(8 x 10-2 )
(2 x1017 )(6 x 105 )
3. Given the density equation D = M/V, rearrange ( a word I will not use again, in
Physics we say “solve for”) the equation to obtain an expression for V.
4. Given the equation for work, W = Fx, and for power, P = W/t, solve for P in terms
of F,x, and t. In other words, your final answer should be an equation for power
with F, x, and t on the other side of the equals sign.
5. Given the following equation: x = vot + ½ gt2, and realizing that vo is equal to
zero, solve for an expression for t, which is time.
6. Provide the correct response:
(a) 5 + 60/3 – 5 =
(b) 22 + 5 =
7. Given an expression for x(t), (read as “position as a function of time”) use the
quadratic formula to solve for time:
x(t) = -4.9t2 + 16t - 8
(in which “x” is measured in meters)
(special note: if you got 2 positive answers that is correct! This particular expression is
describing an object tossed up into the air, so the times correspond to the object on the
way up versus on the way down!)
8. Using the following data, prepare a graph depicting force vs. position:
Force (N)
Position (meters)
N represents the Newton which is the SI unit for force and position is
almost always represented by “x” so let us retitle this data table:
Force (N)
25
35
45
55
65
75
85
95
105
110
x (m)
5.0
7.25
8.5
11.0
13.0
15.25
17.5
19.25
21.0
23.5
Determine the slope of the best fit line. We draw the best fit line as a way
of representing the trend established by the data.
The data presented here represents the force required to stretch a spring;
the relationship is clearly linear and the slope of the line is called the force
constant of the spring.
Record the force constant of the spring with proper units:_____________
9. Given the legs of a right triangle as 20cm and 35 cm, determine the hypotenuse of
the triangle.
10. Given the following displacement triangle depicting two separate displacements
being added together, determine the missing displacement and determine the
angle indicated by the “?.”
10 m
?m
?
8m
OK! Please take the time to present your solutions neatly and on separate paper.
Quadrilled paper (graphing paper) can be downloaded free, so please present the solution
to #8 in an appropriate manner. Make sure you take the time to find another Physics
student to be your buddy so you can compare answers for correctness.
Make sure your work is complete and ready to be turned in on Friday, Sept.4.
I am looking forward to the fall and another year of Physics instruction.
Be seeing you!