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# Category Archives: Physics

### Reasoning

(from Modern Chemistry, Davis, HRW)

### As shown below, the coefficients in a chemical reaction involving gases indicate the relative numbers of molecules, the relative numbers of moles, and the relative volumes. ## As an equation

### or ## Example problems

### V1 / n1 = V2 / n2

2.00 L / 0.500 mol = 2.70 L / x

x = 0.675 mol

### 0.675 mol – 0.500 mol = 0.175 mol

0.175 mol x 4.00 g/mol = 0.7 grams of He added

### We can perform a calculation using Avogadro’s Law:

V1 / n1 = V2 / n2

Let’s assign V1 to be 1 L and V2 will be our unknown.

Let us assign 1 mole for the amount of neon gas and assign it to be n1.

The mass of argon now added is exactly equal to the neon, but argon has a higher gram-atomic weight (molar mass) than neon. Therefore less than 1 mole of Ar will be added. Let us use 1.5 mol for the total moles in the balloon (which will be n2) after the Ar is added. (I picked 1.5 because neon weighs about 20 g/mol and argon weighs about 40 g/mol.)

1 / 1 = x / 1.5

x = 1.5

### V1 / n1 = V2 / n2

 5.120 L 18.10 L –––––––– = –––––– 8.500 mol x

x = 30.05 mol <— total moles, not the moles added

30.05 – 8.500 = 21.55 mol (to four sig figs)

## Charles’s Law

### This relationship can be written as: ### This relationship can be written as: ### Important! This is not a law of physics! Rather, this is a generally useful rule, which is only valid when gas temperature and pressure is low enough for the atoms to usually be far apart from each other.  As we begin to deal with more extreme cases, this rule doesn’t hold up.

Let’s see this in action! https://www.grc.nasa.gov/WWW/K-12/airplane/aglussac.html

## Origin

### Named after Jacques Alexandre César Charles (1746 – 1823)  a French inventor, scientist, mathematician, and balloonist. Just so we’re all clear on this, he was kind of a mad scientist. And I say that with the utmost approval! Contemporary illustration of the first flight by Prof. Jacques Charles with Nicolas-Louis Robert, December 1, 1783. Viewed from the Place de la Concorde to the Tuileries Palace (destroyed in 1871)

.Apps

Charles’s law app

## Learning standards

Massachusetts Science and Technology/Engineering Curriculum Framework

8.MS-PS1-4. Develop a model that describes and predicts changes in particle motion, relative spatial arrangement, temperature, and state of a pure substance when thermal energy is added or removed.

Next Generation Science Standards

MS-PS1-4. Develop a model that predicts and describes changes in particle motion, temperature, and state of a pure substance when thermal energy is added or removed.

College Board Standards

Objective C.1.5 States of Matter

C-PE.1.5.2 Explain why gases expand to fill a container of any size, while liquids flow and spread out to fill the bottom of a container and solids hold their own shape. Justification includes a discussion of particle motion and the attractions between the particles.

C-PE.1.5.3 Investigate the behavior of gases. Investigation is performed in terms of volume (V ), pressure (P ), temperature (T ) and amount of gas (n) by using the ideal gas law both conceptually and mathematically.

Common Core Math

Analyze proportional relationships and use them to solve real-world and mathematical problems.

Ratios & Proportional Relationships

Ratios & Proportional Relationships

CCSS.MATH.CONTENT.7.RP.A.2

Recognize and represent proportional relationships between quantities.

CCSS.MATH.CONTENT.7.RP.A.2.A

Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.

CCSS.MATH.CONTENT.7.RP.A.2.B

Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

## Pascal’s Principle

Pressure applied to an enclosed, incompressible, static fluid is transmitted undiminished to all parts of the fluid.

Hydraulic systems operate according to Pascal’s law.

11.5 Pascal’s Principle

• Define pressure.
• State Pascal’s principle.
• Understand applications of Pascal’s principle.
• Derive relationships between forces in a hydraulic system. image from littlewhitecoats.blogspot.com

.

tba

## Bernoulli’s equation

### Online textbook

Bernoulli’s Equation

The Most General Applications of Bernoulli’s Equation

Viscosity and Laminar Flow; Poiseuille’s Law

The onset of turbulence

Motion of an Object in a Viscous Fluid

Molecular Transport Phenomena: Diffusion, Osmosis, and Related Processes

### Air flow over airplane wings image from http://www.thaitechnics.com

## Resources

NASA How the Bernoulli equation works in rockets

Bernoulli’s Equation OpenStax College

4physics The Bernoulli effect

Energyeducation.ca Bernoulli’s equation

## Apps

PhET fluid pressure and flow

lmnoeng.com Bernoulli equation calculator

Endmemo.com Bernoulli equation calculator

## Learning Standards

2016 Massachusetts Science and Technology/Engineering Curriculum Framework

HS-PS3-1. Use algebraic expressions and the principle of energy conservation to calculate the change in energy of one component of a system when the change in energy of the other component(s) of the system, as well as the total energy of the system including any energy entering or leaving the system, is known.

Disciplinary Core Idea Progression Matrix

PS3.A and 3.B: The total energy within a physical system is conserved. Energy transfer within and between systems can be described and predicted in terms of energy associated with the motion or configuration of particles (objects)

## NGSS leaves out critical guidance on importance of teaching about vectors As we all know the NGSS are more about skills than content. Confusingly, though, they ended up also listing core content topics as well – yet they left out kinematics and vectors, the basic tools needed for physics in the first place.

The NGSS also dropped the ball by often ignoring the relationship of math to physics. They should have noted which math skills are needed to master each particular area.

Hypothetically, they could have had offered options: For each subject, note the math skills that would be needed to do problem solving in this area, for

* a standard (“college prep”) level high school class
* a lower level high school class, perhaps along the lines of what we call “Conceptual Physics” (still has math, but less.)
* the highest level of high school class, the AP Physics level. And the AP study guides already offer what kinds of math one needs to do problem solving in each area.

Yes, the NGSS does have a wonderful introduction to this idea, (quoted below) – but when we look at the actual NGSS standards they don’t mention these skills.

In some school districts this has caused confusion, and even led to some administrators demanding that physics be taught without these essential techniques (i.e. kinematic equations, conceptual understanding of 2D motion, kinematic analysis of 2D motion, vectors, etc.)

To help back up teachers in the field I put together these standards for vectors, from both science and mathematics standards.

– Robert Kaiser

## Learning Standards

Massachusetts Science Curriculum Framework (pre 2016 standards)

1. Motion and Forces: Central Concept: Newton’s laws of motion and gravitation describe and predict the motion of most objects.
1.1 Compare and contrast vector quantities (e.g., displacement, velocity, acceleration force, linear momentum) and scalar quantities (e.g., distance, speed, energy, mass, work).

NGSS

Science and Engineering Practices: Using Mathematics and Computational Thinking

Mathematical and computational thinking in 9–12 builds on K–8 experiences and progresses to using algebraic thinking and analysis, a range of linear and nonlinear functions including trigonometric functions, exponentials and logarithms, and computational tools for statistical analysis to analyze, represent, and model data. Simple computational simulations are created and used based on mathematical models of basic assumptions.

• Apply techniques of algebra and functions to represent and solve scientific and engineering problems.

Although there are differences in how mathematics and computational thinking are applied in science and in engineering, mathematics often brings these two fields together by enabling engineers to apply the mathematical form of scientific theories and by enabling scientists to use powerful information technologies designed by engineers. Both kinds of professionals can thereby accomplish investigations and analyses and build complex models, which might otherwise be out of the question. (NRC Framework, 2012, p. 65)

Students are expected to use mathematics to represent physical variables and their relationships, and to make quantitative predictions. Other applications of mathematics in science and engineering include logic, geometry, and at the highest levels, calculus…. Mathematics is a tool that is key to understanding science. As such, classroom instruction must include critical skills of mathematics. The NGSS displays many of those skills through the performance expectations, but classroom instruction should enhance all of science through the use of quality mathematical and computational thinking.

Common Core Standards for Mathematics (CCSM)

High School: Number and Quantity » Vector & Matrix Quantities. Represent and model with vector quantities.

#### Represent and model with vector quantities.

CCSS.MATH.CONTENT.HSN.VM.A.1
(+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).
CCSS.MATH.CONTENT.HSN.VM.A.2
(+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
CCSS.MATH.CONTENT.HSN.VM.A.3
(+) Solve problems involving velocity and other quantities that can be represented by vectors.

#### Perform operations on vectors.

CCSS.MATH.CONTENT.HSN.VM.B.4
CCSS.MATH.CONTENT.HSN.VM.B.4.A
Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
CCSS.MATH.CONTENT.HSN.VM.B.4.B
Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
CCSS.MATH.CONTENT.HSN.VM.B.4.C
Understand vector subtraction v – w as v + (-w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.
CCSS.MATH.CONTENT.HSN.VM.B.5
(+) Multiply a vector by a scalar.
CCSS.MATH.CONTENT.HSN.VM.B.5.A
Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vxvy) = (cvxcvy).
CCSS.MATH.CONTENT.HSN.VM.B.5.B
Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).
American Association for the Advancement of Science

Operating with Symbols and Equations

• Become fluent in generating equivalent expressions for simple algebraic expressions and in solving linear equations and inequalities.
• Develop fluency operating on polynomials, vectors, and matrices using by-hand operations for the simple cases and using technology for more complex cases.
AAAS Benchmarks (American Association for the Advancement of Science)
9B9-12#2: Symbolic statements can be manipulated by rules of mathematical logic to produce other statements of the same relationship, which may show some interesting aspect more clearly. Symbolic statements can be combined to look for values of variables that will satisfy all of them at the same time.
9B9-12#5: When a relationship is represented in symbols, numbers can be substituted for all but one of the symbols and the possible value of the remaining symbol computed. Sometimes the relationship may be satisfied by one value, sometimes more than one, and sometimes maybe not at all.
12B9-12#2: Find answers to problems by substituting numerical values in simple algebraic formulas and judge whether the answer is reasonable by reviewing the process and checking against typical values.
12B9-12#3: Make up and write out simple algorithms for solving problems that take several steps.

#vectors #teaching #standards #kinematics #physics #kaiserscience #pedagogy #education #NGSS #Benchmarks #scalors #highschool

## Two and three dimensional motion

Most high school physics courses don’t include algebraic analysis of two or three dimensional kinematics and momentum. But these clearly are of great importance.

In a regular, college prep physics high school setting, I can’t imagine skipping 2D physics! Even if we don’t do 2D kinematic equations, we need to cover 2D vectors, and show examples of parabolic motion, and vector components.

https://www.physicsclassroom.com/class/vectors/Lesson-1/Vector-Components

and

https://www.physicsclassroom.com/class/vectors/Lesson-2/What-is-a-Projectile

Even if we are not doing the math, I want them to see examples of conservation of momentum in two dimensions, like this:

https://www.physicsclassroom.com/mmedia/momentum/2di.cfm image from physicsclassroom

These seem like great apps for teaching 2D kinematics without all of the detailed calculations.

https://www.physicsclassroom.com/Physics-Interactives/Vectors-and-Projectiles

We won’t be able to do three dimensional collision or momentum problem solving, but we can at least introduce the idea of it, and show them why almost every collision and conservation of momentum in the real world is 3D:

First students need to be introduced to the idea that there are more than just two axes (X and Y,) there is a third dimension, Z! Then we see how we can plot points in three dimensions.

This is the GeoGebra app

Now we realize that the size and motion of any object can be plotted in three dimensions.

The physics of dogs and cats colliding GIF Two galaxies colliding, and the resulting amazing display

https://media2.giphy.com/media/T2C2efftbhMTC/giphy.webp?cid=790b7611ec6f62c760e4a1059d923e3f271feb87182a154f&rid=giphy.webp

A practical use of 2D kinematics and conservation of momentum: forensic accident reconstruction. ## Boyle’s law (gas laws)

A general relationship between pressure and volume: Boyle’s Law

As the pressure on a gas increases, the volume of the gas decreases because the gas particles are forced closer together.

Conversely, as the pressure on a gas decreases, the gas volume increases because the gas particles can now move farther apart.

Example: Weather balloons get larger as they rise through the atmosphere to regions of lower pressure because the volume of the gas has increased; that is, the atmospheric gas exerts less pressure on the surface of the balloon, so the interior gas expands until the internal and external pressures are equal.

from Libretexts, Chemistry, 5.3: The Simple Gas Laws: Boyle’s Law, Charles’s Law and Avogadro’s Law, CC BY-NC-SA 3.0.

This means that, at constant temperature, the pressure (P) of a gas is inversely proportional to the volume (V).

PV = c

Important! This is not a law of physics! Rather, this is a generally useful rule, which is only valid when gas temperature and pressure is low enough for the atoms to usually be far apart from each other.  As we begin to deal with more extreme cases, this rule doesn’t hold up.

Let’s see the relationship in action, here: Early scientists explored the relationships among the pressure of a gas (P) and its temperature (T), volume (V), and amount (n) by holding two of the four variables constant (amount and temperature, for example), varying a third (such as pressure), and measuring the effect of the change on the fourth (in this case, volume).

The history of their discoveries provides several excellent examples of the scientific method.

The Irish chemist Robert Boyle (1627–1691) carried out some of the earliest experiments that determined the quantitative relationship between the pressure and the volume of a gas. Boyle used a J-shaped tube partially filled with mercury.

In these experiments, a small amount of a gas or air is trapped above the mercury column, and its volume is measured at atmospheric pressure and constant temperature. More mercury is then poured into the open arm to increase the pressure on the gas sample.

The pressure on the gas is atmospheric pressure plus the difference in the heights of the mercury columns, and the resulting volume is measured. This process is repeated until either there is no more room in the open arm or the volume of the gas is too small to be measured accurately. Details: Boyle’s Experiment Using a J-Shaped Tube to Determine the Relationship between Gas Pressure and Volume.

(a) Initially the gas is at a pressure of 1 atm = 760 mmHg (the mercury is at the same height in both the arm containing the sample and the arm open to the atmosphere); its volume is V.

(b) If enough mercury is added to the right side to give a difference in height of 760 mmHg between the two arms, the pressure of the gas is 760 mmHg (atmospheric pressure) + 760 mmHg = 1520 mmHg and the volume is V/2.

(c) If an additional 760 mmHg is added to the column on the right, the total pressure on the gas increases to 2280 mmHg, and the volume of the gas decreases to V/3

(This section from from Libretexts, Chemistry, 5.3: The Simple Gas Laws: Boyle’s Law, Charles’s Law and Avogadro’s Law, CC BY-NC-SA 3.0)

## Learning standards

Massachusetts Science and Technology/Engineering Curriculum Framework

8.MS-PS1-4. Develop a model that describes and predicts changes in particle motion, relative spatial arrangement, temperature, and state of a pure substance when thermal energy is added or removed.

Next Generation Science Standards

MS-PS1-4. Develop a model that predicts and describes changes in particle motion, temperature, and state of a pure substance when thermal energy is added or removed.

College Board Standards

Objective C.1.5 States of Matter

C-PE.1.5.2 Explain why gases expand to fill a container of any size, while liquids flow and spread out to fill the bottom of a container and solids hold their own shape. Justification includes a discussion of particle motion and the attractions between the particles.

C-PE.1.5.3 Investigate the behavior of gases. Investigation is performed in terms of volume (V ), pressure (P ), temperature (T ) and amount of gas (n) by using the ideal gas law both conceptually and mathematically.

Common Core Math

Analyze proportional relationships and use them to solve real-world and mathematical problems.

Ratios & Proportional Relationships

Ratios & Proportional Relationships

CCSS.MATH.CONTENT.7.RP.A.2

Recognize and represent proportional relationships between quantities.

CCSS.MATH.CONTENT.7.RP.A.2.A

Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.

CCSS.MATH.CONTENT.7.RP.A.2.B

Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.