Unit 1. day 6
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This document discusses reflections in geometry. It begins with essential questions about reflections and how they differ from translations. It then discusses properties of reflections, including that lines are taken to lines of the same length and angles are taken to angles of the same measure. It provides examples of reflecting objects across different lines of reflection, including non-standard lines like y=3 and x=-2. It includes an example of two students, Antonio and Brittany, where Brittany tries to mirror Antonio's movements on a coordinate plane.
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NORTHERN ILLINOIS UNIVERSITY
PHYSICS DEPARTMENT
Physics 253 – Basic Mechanics Fall 2016
Problem Set #1
Problem Set Due: Wed., Aug. 29, 2016
Read Lecture Notes #1 & #2 (Giancoli: Chapters 1&2)
Remember to write all answers with the proper # of significant figures
Show explicitly all your work for full credit
Section 2.2:
1. Express the position vector, �⃗� , in terms of (i) unit vectors, and (ii) magnitude and
angle for the two cases below:
(a) Point 𝑃 is located at the coordinate (−3.0,4.0).
(b) Point 𝑃 is located at the coordinate (-1.25,-4.6).
Make certain you express your solutions with the proper # of significant figures.
𝑥
𝑦
𝑟
𝑃
𝑥
𝑦
𝑃
𝑟
Section 2.3:
2. An object’s position changes with time according to the graph shown below. For the
positions located at Points A, B, C, state whether the velocity is positive, negative, or
zero by drawing the tangent lines to the curve and examining their slopes.
(a) At which point is the object moving the slowest?
(b) At which point is the object moving the fastest?
(c) At points A, B, C, state whether the velocity is positive, negative, or zero by
drawing the tangent lines to the curve and examining their slopes.
𝑥
𝑡
𝐴 (𝑎𝑡 𝑡ℎ𝑒 𝑣𝑒𝑟𝑦 𝑡𝑜𝑝 𝑜𝑓 𝑝𝑒𝑎𝑘)
𝐵
𝐶
3. The derivative of a polynomial function,
n
t , is defined as:
1
n
ndt
nt
dt
Example: The position of a particle at time t is given by 37.8 9.2 2.1x t t t
Its velocity will then be
0 1 1 1 3 1
0 7.8 1 9.2 3 2.1
dx
v t t t
dt
2
0 9.2 6.3t
(a) Find the derivative of
2 3 4
0 1 2 3 4 x a at a t a t a t where 0a , 1a , 2a , 3a ,
and 4a are constants.
(b) Find the derivative of
1 2 3 4
0 2 3 4
a a a a
x a
t t t t
where 0a , 1a , 2a , 3a , and 4a
are constants.
(c) What is the derivative of x t ?
Section 2.4:
4. The position of a particle at time t is given by 210 0 2x t x v t at where 0x , 0v
, and a are constants independent of time.
(a) What is the velocity of the particle at time t ?
(b) What is the acceleration of the particle at time t ?
5. The position of a particle moving on the x-axis is given by
. 𝑥 = (6.5𝑚) + (3.7𝑚/𝑠𝑒𝑐)𝑡 − (0.2𝑚/𝑠𝑒𝑐2)𝑡2
with x in meters and t in seconds.
(a) What is the velocity of the particle at 𝑡 = 2.6 sec?
(b) Is the velocity constant or does it change with time?
Section 2.5:
The integral
Chapter 3. vectors projectiles
This document provides information about two-dimensional motion and vectors. It introduces vectors, including that vectors have both magnitude and direction while scalars only have magnitude. It discusses basic vector operations like addition and subtraction of vectors. It also provides examples of calculating the displacement and direction of objects moving in two dimensions using vector addition. Key concepts covered include resolving vectors into horizontal and vertical components, and that the horizontal and vertical motions of a projectile are independent.
Arts revealed in calculus and its extension
Motivated by presenting mathematics visually and interestingly to common people based on calculus and its extension, parametric curves are explored here to have two and three dimensional objects such that these objects can be used for demonstrating mathematics.
Epicycloid, hypocycloid are particular curves that are implemented in MATLAB programs and the motifs are presented here. The obtained curves are considered to be domains for complex mappings to have new variation of Figures and objects. Additionally Voronoi mapping is also implemented to some parametric curves and some resulting complex mappings.
Some obtained 3 dimensional objects are considered as flowers and animals inspiring to be mathematical ornaments of hypocycloid dance which is also illustrated here.
Transformational geometry
Transformations are geometric operations that change the position of a shape in space. The document defines and explains three common transformations: translation, rotation, and reflection. Translation moves a shape along a straight path without changing its orientation. Rotation turns a shape around a fixed point. Reflection produces a mirror image of a shape across a line or plane. The document provides step-by-step instructions for performing each type of transformation on a shape. It also includes an example problem demonstrating multiple consecutive transformations.
Foundation in engineering and technologyv1 copy(1)
This document provides an overview of foundation and materials science studies, including physics and mathematics. It covers topics such as physical quantities and units, scalars and vectors, and multiplication of vectors. Specifically, it defines basic and derived physical quantities and their SI units. It also describes vector addition and subtraction graphically using parallelograms and triangles. Vector components are resolved into x- and y-axes and unit vectors. Scalar (dot) and vector (cross) products are defined, with scalar products providing the parallel component between two vectors and vector products determining the perpendicular component. Formulas for various physical quantities like work and momentum are also presented.
Foundation in engineering and technologyv1
This document provides an overview of foundation and materials science studies, including physics and mathematics. It covers topics such as physical quantities and units, scalars and vectors, and multiplication of vectors. Specifically, it defines basic and derived physical quantities and their SI units. It also describes vector addition and subtraction graphically using parallelograms and triangles. Vector components are resolved into x- and y-axes and unit vectors. Scalar (dot) and vector (cross) products are defined, with scalar products providing the parallel component between two vectors and vector products determining the perpendicular component. Examples of each are given using unit vectors.
Combinations
This document provides an overview of isometries and their composition. It outlines objectives for students to classify isometries and find compositions, references relevant Iowa Common Core standards, defines isometries as transformations that preserve distance, describes the four kinds of isometries including reflections, rotations, translations, and glide reflections, and provides examples of reflections over parallel and intersecting lines as well as glide reflections using Sketchpad. Homework assignments are included.
(8) Inquiry Lab - Composition of Transformations
A combination of transformations differs from a single transformation in that it involves performing multiple transformations in sequence, while a single transformation only involves one type of transformation. They are the same in that both transformations preserve properties like size, shape, and angle measurements of the original figure. The document provides examples of using reflections and translations together to transform shapes and create decorative borders, demonstrating how combinations of transformations can be used.
Unit 1. day 5. quiz 1
This document contains instructions and content for a math quiz on geometry topics including translations, rotations, reflections, and dilations. It begins with a warm-up involving determining if figures are similar and solving an equation about translating a figure. Then it lists the common core standards addressed, poses an essential question about translating a figure, previews the warm-up quiz questions, and ends by providing time for the quiz and collecting it.
Lecture 1.ppt
The document discusses fundamental concepts in measurements and physical quantities. It defines key terms like measurement, physical quantity, fundamental quantities, derived quantity, scalars, vectors and units. Four fundamental quantities are identified: length, mass, time and electric charge. Other concepts covered include the International System of Units (SI), measurement errors, accuracy and precision. Properties of vectors such as addition and representation are also explained.
11 - 3 Experiment 11 Simple Harmonic Motio.docx
11 - 3
Experiment 11
Simple Harmonic Motion
Questions
How are swinging pendulums and masses on springs related? Why are these types of
problems so important in Physics? What is a spring’s force constant and how can you measure
it? What is linear regression? How do you use graphs to ascertain physical meaning from
equations? Again, how do you compare two numbers, which have errors?
Note: This week all students must write a very brief lab report during the lab period. It is
due at the end of the period. The explanation of the equations used, the introduction and the
conclusion are not necessary this week. The discussion section can be as little as three sentences
commenting on whether the two measurements of the spring constant are equivalent given the
propagated errors. This mini-lab report will be graded out of 50 points
Concept
When an object (of mass m) is suspended from the end of a spring, the spring will stretch
a distance x and the mass will come to equilibrium when the tension F in the spring balances the
weight of the body, when F = - kx = mg. This is known as Hooke's Law. k is the force constant
of the spring, and its units are Newtons / meter. This is the basis for Part 1.
In Part 2 the object hanging from the spring is allowed to oscillate after being displaced
down from its equilibrium position a distance -x. In this situation, Newton's Second Law gives
for the acceleration of the mass:
Fnet = m a or
The force of gravity can be omitted from this analysis because it only serves to move the
equilibrium position and doesn’t affect the oscillations. Acceleration is the second time-
derivative of x, so this last equation is a differential equation.
To solve: we make an educated guess:
Here A and w are constants yet to be determined. At t = 0 this solution gives x(t=0) = A,
which indicates that A is the initial distance the spring stretches before it oscillates. If friction is
negligible, the mass will continue to oscillate with amplitude A. Now, does this guess actually
solve the (differential) equation? A second time-derivative gives:
Comparing this equation to the original differential equation, the correct solution was
chosen if w2 = k / m. To understand w, consider the first derivative of the solution:
−kx = ma
a = −
k
m
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
x
d 2x
dt 2
= −
k
m
x x(t) = A cos(ωt)
d 2x(t)
dt 2
= −Aω2 cos(ωt) = −ω2x(t)
James Gering
Florida Institute of Technology
11 - 4
Integrating gives
We assume the object completes one oscillation in a certain period of time, T. This helps
set the limits of integration. Initially, we pull the object a distance A from equilibrium and
release it. So at t = 0 and x = A. (one.
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Unit 1. day 16
This document provides information about different types of triangles, parallel lines cut by a transversal, and finding missing angle measurements. It includes examples of acute, obtuse, right, scalene, isosceles, and equilateral triangles. It also discusses same-side interior angles, same-side exterior angles, transversals, alternate interior angles, and linear pairs related to parallel lines cut by a transversal. The document poses questions about finding specific angle measurements and the value of x given parallel lines and a transversal. It provides a word problem involving replacing a broken window where only two angle measurements are given and asks how to find the other missing angle measurements.
Unit 1. day 15b
This document contains information about angle relationships formed when parallel lines are cut by a transversal line. It defines terms like alternate interior angles, linear pairs, same-side interior angles, and same-side exterior angles. It includes examples of labeling angles in different geometric figures and problems involving solving for the value of x. The closing asks students to reflect on what they learned about parallel lines and transversals and any struggles they had with the concepts.
Unit 1. day 15
This document discusses parallel lines crossed by a transversal and the special angles that are formed - alternate interior angles, same-side interior angles, same-side exterior angles, and linear pairs. It provides examples of labeling these angles and finding their measurements when the lines are parallel. The final section describes a word problem where students must label angles, find their measurements, and determine if lines appear to be parallel based on the angle measurements.
Unit 1. day 14
This document discusses angle relationships when parallel lines are cut by a transversal. It defines key terms like alternate interior angles, corresponding angles, and vertical angles. It explains that alternate interior angles, corresponding angles, and vertical angles are congruent, while same-side interior and same-side exterior angles are supplementary. The document provides examples of solving for missing angle measures using these properties and asks students to identify angle relationships in diagrams.
Unit 1. day 13
This document contains instructions for an activity where students will work in pairs to recreate geometric transformations by describing them to their partner without showing the example card. The activity focuses on transformations including rotations, reflections, translations, and dilations. Students are provided 20 example cards and must use precise language around geometric concepts like vertices, corresponding sides/angles, and coordinates when explaining each transformation to their partner.
Unit 1. day 12
This document provides instruction on dilations including definitions, examples, and practice problems. It begins with warm-up math problems, then defines a dilation as a transformation where the scale factor multiplies the coordinates. Examples show plotting points and dilating a triangle by scaling the coordinates. Practice problems involve dilating triangles on a coordinate plane using given scale factors and solving for the new vertex coordinates.
Unit 1. day 11
This document provides information about dilations including:
1) Dilations change the size of a figure but not its shape, resulting in similar figures related by a scale factor.
2) A scale factor greater than 1 means the dilated figure is larger than the original, while a scale factor less than 1 means it is smaller.
3) Examples are given of dilating a figure using a scale factor and finding the scale factor from coordinate points.
Unit 1. day 7
The document provides instructions on how to perform rotations on a coordinate plane, explaining that a rotation involves turning a figure around a fixed point by a certain degree, such as 90 or 180 degrees, and examples are given of rotating simple figures and points around the origin. Rules are given for rotating points 90 and 180 degrees clockwise and counterclockwise around the origin by changing the x and y coordinates. Students are asked to graph rotated images and describe transformations.
Unit 1. day 3review
This document contains a math review on translations, similar and congruent figures, and coordinates. It includes warm-up problems, essential questions about translations and transformations, examples of translating figures and finding new coordinates, work on classifying figures as similar or congruent and identifying corresponding parts, and examples of translating points and finding new coordinates. The key ideas are that translations move figures by sliding them along parallel lines without changing their shape or size, and correspond to algebraic expressions of (x + a), (y + b) that shift the x- and y-coordinates.
Unit 1. day 11b
This document provides information about similar and congruent figures including:
1) Definitions of corresponding sides and angles, congruent figures, and similar figures. Similar figures have the same shape but not necessarily the same size, while congruent figures have the same size and shape.
2) Examples are given to identify the scale factor of similar figures as the ratio of corresponding sides.
3) Worked examples ask students to identify corresponding sides and angles of similar triangles, and use scale factors to find missing side lengths.
4) A closing asks students to share answers and discuss as needed.
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Unit 1. day 6
- 1. Warm-Up Determine the scale factor of the similar polygons. 1. 2.
- 2. Essential Question How is a translation different from a reflection?
- 3. Reflections Day 2 Unit 1, Day 6
- 4. Common Core GPS MCC8.G.1: Verify experimentally the properties of rotations, reflections, and translations: a. Lines are taken to lines, and line segments to line segments of the same length. b. Angles are taken to angles of the same measure. c. Parallel lines are taken to parallel lines. MCC8. G. 2: Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
- 5. Language of the Standards What is a reflection? What is a reflection line?
- 6. Lines of Reflection Not all reflection lines include the x and y-axis. We may also reflect an object using a line that is shifted on the axis such as y = 3 and x = -2. Will this shift in the line of reflection change the shape or size of the object? How do you know?
- 7. Reflect Let’s try a reflection over y = 1.
- 8. Reflect Let’s reflect the same object at x = 1.
- 9. What is the line of reflection? Name the line of reflection.
- 10. What is the line of reflection? Name the line of reflection.
- 11. What is the line of reflection? Name the line of reflection.
- 12. Work Session: Name the line of reflection.
- 13. Graph the figure of the image after the given transformation.
- 15. Antonio and his friend, Brittany, were at a summer math camp that had a large coordinate plane drawn on the gym floor. Antonio challenged Brittany to try and mirror him as he traveled around the first quadrant. Map Antonio’s and Brittany’s movements on this coordinate plane. Antonio began at (2, 1) and walked to (3, 5); Brittany decided to begin at (-2, 1), then tried to mirror Antonio by walking to (-3, 5). Antonio jumped to (5, 5) and side-stepped to (4, 3); Brittany jumped to (-5, 5) and side-stepped to (-4, 3). Antonio returned to (2, 1) and Brittany returned to (-2, 1).
- 16. 7. Did Brittany mirror Antonio? _________________ 8. If Brittany had instead begun at (-2, 1), walked to (-4, 3), side-stepped to (-5, 5), jumped to (-3, 5) and then returned to (-2, 1), could she claim that she created a mirror image of Antonio’s path? Justify your answer.
- 17. Closing: What have you learned about reflections?