Chapter 2 Part 3 Four Kinematic Equations

Prof. Clements Notes/Hints for Physics Courses

Chapter 2 Open Stax College Physics or Most Introductory Physics Courses

Motion Equations, Kinematics

I highly recommend that you go to the web site https://openstax.org/details/books/college-physics to access the free OpenStax College Physics textbook and resources.

YouTube videos of my (usually short) lectures and example problems are indexed at http://www.physics.gpclements.com/ . There are also a few videos that give review of basic math tools that are used in an introductory physics course.

Terms to know: constant acceleration, kinematic equations

Motion Equations The early part of this chapter in most textbooks describes position, velocity, and acceleration. Now these quantities will be connected in equations. There are four equations of motion that we will use. You should be familiar with the book’s notation and my notation. There may be some slight differences in our notation. The time value at the start of the interval will be taken to be zero. The final time is labeled t. Any symbol with a subscript 0 is a quantity at the start of the time interval (often time = 0). The quantities without a subscript are the final values.

If the value of the acceleration changes during a problem then you must work the problem in separate steps, with the acceleration constant during each step.

If the acceleration is constant the acceleration may be some positive number or zero or some negative numbers. “Constant acceleration” does not automatically mean that the acceleration is zero. Also, some students have a misconception that if the acceleration is constant then the value of the velocity is zero. The previous statement is not true.

If the acceleration is a constant +5 m/s² , then every second the velocity will increases by 5 m/s. i.e. Suppose the velocity is +3 m/s at time = 0. After one second the velocity would be +8 m/s. After two seconds the velocity would be +13 m/s. As a check, calculate the average acceleration a = (V – V_o ) / t using the t=0 and t=2 seconds data.

There are two basic equations of motion. Using these two equations we can apply some algebra and develop two more useful equations. There are four useful kinematic equations of motion. The term kinematic tells us that we are not worrying about the cause of motion with these equations. The kinematic equations just describe motion.

The following development of the four equations may be in a slightly different order than your textbook but we will end up with the same equations of motion.

1^st Kinematic Equation: By definition a = V / t or a = (V - V_o ) / t Multiply both sides by t and add V_o to both sides.

Equation 1 V = V_o + at

Suppose the acceleration is 4 m/s², the initial velocity is +3 m/s, and the initial position is 7 meters. What is the velocity when t = 5 seconds? Answer: V = V_o + at

V = 3 m/s + 4 m/s² * 5 sec V = 23 m/s

2^nd Kinematic Equation: The next equation lets us calculate the final position if we know the initial position, the average velocity and the time. For the case of constant acceleration It turns out that the average velocity can be found by adding the initial and final velocities and then dividing by 2. The displacement is equal to the average velocity multiplied by time: X - X_o = [ ½ (V + V_o) ] * t or

X = X_o + [ ½ (V + V_o) ] * t I will refer to this as Equation 2.

Your textbook may use this form S = [ ½ (V + V_o) ] * t Here S = X - X_o

This equation can also be derived by analyzing the graph of V vs t as shown in this image.

X = X_o + ½ (V + V_o) t    This is only valid when the acceleration is constant because the average velocity will not equal ½ (V + V_o) if the acceleration is not constant.

Let us continue to use the data given in the example calculation done for Kinematic Equation 1: X_o = 7 meters V_o = 3 m/s V = 23 m/s a = 4 m/s²

What is the position of the object at t = 5 seconds? Answer: X = X_o + ½ (V + V_o) t

X = 7 meters + 0.5 ( 3m/s + 23 m/s) 5 seconds

X = 7 meters + 0.5 ( 26 m/s) 5 seconds

X = 7 meters + 13 m/s *5 seconds Note * indicates multiplication

X = 7 meters + 65 meters X = 72 meters

3^rd Kinematic Equation: We can combine the first two equations by substituting for V in equation 2 with the expression for V from equation 1. This will generate an equation that does not contain the final velocity. Start with equation 2 X = X_o + ½ (V + V_o) * t

Replace V with the expression from equation 1 X = X_o + ½ ( [V_o + at] + V_o) * t

You should use the principles of algebra to simplify this. Look at the image at the end of this document for help. And, ask your instructor if you don’t obtain:

Equation 3 X = X_o + V_o * t + 0.5 * a * t² or S = V_o * t + 0.5 * a * t²

e.g. Suppose the acceleration is 4 m/s², the initial velocity is +3 m/s, and the initial position is 7 meters. Find the position at time = 5 seconds. Treat the final velocity as unknown.

X = 7 m + 3 m/s * 5 sec + 0.5 * 4 m/s² * ( 5 sec )²

X = 7 m + 15 m + 50 m

X = 72 m Does this result sound familiar?

It is worth examining equation 3 for the case of acceleration = 0. With a = 0 the equation becomes X = X₀ + V₀ * t or (X - X₀ ) = V₀ * t You may recognize this as being similar to your math problems involving Distance = rate * time. D = rt

D = rt is only valid if the rate is constant. (i.e. acceleration = 0, and V = V₀ )

e.g. Initial position = 0 meters, a = 0, initial velocity +4 m/s, time = 3 seconds

D = 4 m/s * 3 seconds = 12 meters

4^th Kinematic Equation: We can combine the first two equations by solving for time in equation 1 and substituting this expression for t into equation 2. This is done in the image at the end of this document.

Equation 4 V² = V_o² + 2 * a * (X – X_o) or V² = V_o² + 2 * a * S

e.g. Suppose the acceleration is 4 m/s², the initial velocity is +3 m/s, the initial position is 7 meters, and the final position is 72 meters. Find the final velocity. Treat time as unknown.

V² = (3 m/s)² + 2 * 4 m/s² * ( 72 meters – 7 meters)

V² = 9 m²/s² + 8 m/s² * 65 m

V² = 9 m²/s² + 520 m²/s² or V² = 529 m²/s²

Take the square root of both sides and you find V = 23 m/s . You should compare this result to the final velocity calculated above for the example for Kinematic Equation 1.

You now know the 4 Kinematic Equations. These equations only give correct results when the acceleration is constant but that covers many situations that you will study in your physics class. The law of Conservation of Energy (future chapter) will help us solve problems when the acceleration is not constant.

You should work through some of the examples in your textbook until you are comfortable using these equations. If there is an example that doesn’t make sense ask your instructor to go over the calculations steps.

Problem-Solving for One-Dimensional Kinematics

Read through the suggested steps for problem solving in your textbook. I strongly recommend that you start by reading the entire problem. Then read it again and make a sketch of the situation. The sketch does not need to be artistic. Sketch the objects (stick figures, boxes, circles, etc.) and select some coordinate system. Label 0 on your coordinate system. Use arrows to indicate velocities and accelerations of objects. Write down the numeric values next to the symbols (X_o, X, V_o , V, a, t ) used in the 4 kinematic equations. For simpler problems you may be able to select one of the four kinematic equations such that there is only one unknown.

For some problems you may have to use more than one equation before you arrive at the final answer.

It is also very important to check that your answer is reasonable. For some situations you won’t know what is reasonable (e.g. speed of an electron). Do your work by yourself and then check with a class mate to see if they have the same answer. Ask your instructor for guidance on rounding or not rounding intermediate values in the calculations. I would suggest that you store intermediate results in the memory of your calculator and don’t round off numbers before you reach the final result. Ask your instructor if you should follow the rules of significant figures for homework and exam problems. In the absence of directions on this I would suggest you write down three digits in your answer. e.g. 5.26 x 10³ m/s , 4.68 m/s² , etc.

The following image shows the derivation of the 4 Kinematic Equations:

Copyright© 2017 by Greg Clements Permission is granted to reproduce this document as long as 1) this copyright notice is included, 2) no charge of any kind is made, and, 3) the use is for an educational purpose. Editing of the document to suit your own class style and purposes is allowed.

Chapter 2 Part 2 Using the graph of position vs time to generate graphs of velocity and acceleration

The included image

shows two examples of the velocity and acceleration graphs that result for given graphs of position versus time.

The production of these graphs is discussed in the first blog for Chapter 2, Displacement, Vectors, Velocity, Acceleration.

Chapter 2 Part 1 Displacement, Vectors, Speed, Velocity, Acceleration

Prof. Clements Notes/Hints for Physics Courses

Chapter 2 Open Stax College Physics or Most Introductory Physics Courses

Part 1: Displacement, Vectors, Speed, Velocity, Acceleration

I highly recommend that you go to the web site https://openstax.org/details/books/college-physics to access the free OpenStax College Physics textbook and resources.

YouTube videos of my (usually short) lectures and example problems are indexed at http://www.physics.gpclements.com/ . There are also a few videos that give review of basic math tools that are used in an introductory physics course.

Terms you should know after this unit: displacement, delta, scalar, vector, speed, velocity, acceleration, average, instantaneous

The study of motion will occupy a significant portion of the first semester of a physics class. This chapter presents the description of motion. Later chapters will present the cause of motion (force). It is important to have a good foundation on the description of motion so you can better understand situations where the force needs to be analyzed. A simplification in this chapter is that we will only let objects move in one line. The object may move back and forth, or up and down, but the motion will be in a straight line.

Displacement

In order to describe the motion of an object we need to make measurements of the position as a function of time. Position measurements are made in a reference frame. A straight line with a zero mark at some location can be a reference frame.

Displacement is the change in position of an object. When working in one dimension the direction of change is indicated by a + or - in front of the size of the displacement.  X = X_f - X_o where X_f is the final position and X_o is the initial position. The  (delta) symbol tells us to subtract two numbers. You only need the final and initial position values. We do not need to keep track of the intermediate positions of the object.

A certain person walks 80 cm North, 60 cm South and 30 cm North. What is the displacement?

Answer:size of the displacement: 50 cm direction of the displacement North

Note that displacement answer is incomplete if it is missing either the size (magnitude) of the displacement or the direction.

Along our X axis we will use + to indicate displacement in the positive direction. A - sign will indicate displacement in the negative direction.

In the real world objects seldom travel on a perfectly straight line with no reversals of motion. As real objects move they will move along curves and straight lines and may back up. The total path length covered is called the distance traveled.

How does distance differ from displacement?

Distance traveled is usually larger than the magnitude of the displacement. Remember, displacement is measured in a straight line.

Distance traveled can be equal to the magnitude of the displacement if the motion is in a straight line with no reversal of motion.

Vectors, Scalars, Coordinate Systems

Quantities that have both a size (magnitude) and direction are called vectors. Displacement is a vector. The + or - sign is enough to indicate direction. You don't have to use words such as left or right to describe the direction of the displacement.

Examples of vectors: displacement, velocity, acceleration, force, electric field …

Examples of scalars: temperature, mass, speed ...

The main difference between a scalar and a vector is that the scalar does not have a direction.

You will be using the Cartesian coordinate system (three axes at right angles to each other, X Y Z). Usually discussions are restricted to motion in at most 2 dimensions. For horizontal motion I will describe positions along a X axis. For vertical motion I will describe positions along a Y axis.

As you work on problems you must choose the positive direction and the negative direction. You should label this direction with a + on the sketch you make for the problem. Once you select the + direction you must make sure all of the values you write down from the word problem are consistent with the + direction you have selected. i.e. If you choose the + direction to the right and the object moves to the right 5 meters in 2 seconds you would use a displacement value of + 5 meters.

You will often have some freedom in where to place the origin of the coordinate system. The placement of the origin does not affect the results of motion calculations. i.e. For an object moving on a straight track it doesn't matter if we choose the origin (X = 0) of the coordinate system to be at the left edge of a track or at the middle of the track or at the right edge of the track. The position numbers do change, but the important quantities of velocity and acceleration will be the same regardless of the location of the origin of the coordinate system.

Time, Velocity, Speed

When time has elapsed there will be a change. e.g. the hands of clock have moved, your heart has made a beat, a leaf has moved, a sound has been detected, the Earth has moved in its orbit around the Sun, etc. The time for motion to take place will be used in calculating velocity and speed. We will often refer to some clock and say the time on the clock is 0 seconds at the start of the motion and equal to some value, t, at the end of the motion. The time interval, Δt = t_f -t_o, will simplify to just t_f when

t₀ has a value of 0.

The average velocity is found by dividing displacement by time. Because displacement is a vector the velocity is also a vector. If an object has a displacement of +8 meters in 4 seconds the average velocity is +2 meters/second. The “+” sign indicates the direction is to the right on the axis.

Instantaneous velocity, v, is the velocity of the object at a particular instant of time.

Consider a car stopped a red light on a city street. Suppose that after the light turns green the driver presses lightly on the gas pedal for two seconds and the car starts moving faster and faster until it reaches some final velocity. During this two second time interval after the light turns green the car will have some average velocity (that will be smaller than the final velocity). The instantaneous velocity will be different at each instant of time until the car reaches its final velocity. The speedometer of the car (approximately) gives the velocity at each instant of time.

Instantaneous velocity, v, can be found by calculating  X /  t as  t approaches 0. This requires the use of Calculus.

Another term that describes motion is speed. Speed and velocity are usually much different quantities. The distinction in these quantities comes from using the distance traveled or the displacement.

The average speed = (total distance traveled) / (time required)

Speed is a scalar because it does not have a direction.

One can also calculate the instantaneous speed by considering a very small time interval.

How far did you drive to come to your school to class?

How much time was required for the trip?

Calculate your average speed distance/time

Imagine a map of your journey and estimate the displacement. Calculate your average velocity.

displacement/time → ______________ ____________

(The first blank is for the magnitude of the average velocity and the second blank is for the direction.)

The average velocity value usually be smaller than the average speed. This occurs because the straight line displacement magnitude is usually smaller than the distance traveled (that includes curves, etc.)

The size of the instantaneous velocity is called the speed of the object. Speed does not have a direction. Quantities that do not have an associated direction are called scalars. Speed is a scalar.

Acceleration

The measure of the rate of change of velocity is called acceleration. The average acceleration is found by dividing the quantities v and t.

a_avg = v/ t The direction of the acceleration should be specified, but this can be done with + and - .

Suppose an object has a velocity of + 2 m/s at time = 0 seconds and then has a velocity of +9 m/s at a time of 2 seconds. What is the average acceleration?

Answer: The average acceleration is ( 9 m/s - 2 m/s) / 2 seconds or (7 m/s) / 2s or +3.5 m/s²

The meaning of an acceleration of +3.5 m/s² is that the velocity is increasing by 3.5 m/s every second. If the acceleration value is constant, the velocity would be 2 m/s at time 0, 5.5 m/s at time = 1 second, and 9 m/s at time = 2 seconds. If the acceleration continues the velocity would be 12.5 m/s at time = 3 seconds.

Some texts discuss deceleration. The key concept is that if the acceleration has a - (negative) value then the velocity is becoming more negative each second. Acceleration values will be + and - in homework and exam problems. e.g. At time = 0 the velocity is 15 m/s and the acceleration is -4 m/s². At time = 1 second the velocity would be 11 m/s. At time = 2 seconds the velocity would be 7 m/s, etc.

Note that the mathematical procedure used to calculate average velocity is the same as the procedure used for calculating average acceleration: a difference is divided by a time interval.

The instantaneous acceleration is the acceleration value at an instant in time.

An estimate can be made for the instantaneous velocity at some desired time, t, from the graph of position vs time. The method involves: 1) Draw a straight line that grazes the graph of position vs time at the desired time, t. This line is often called a “tangent” line. On each side of the place where the straight line touches the graph of position vs time, there will be equal gaps between the straight line and the graph of position vs time. See your textbook for examples. 2) Calculate the slope of the straight line (rise/run). This is the instantaneous velocity.

A similar procedure for the graph of velocity vs time will yield values for the instantaneous acceleration.

These calculations for instantaneous velocity and acceleration will be approximate as you cannot draw a “perfect” tangent line, nor calculate the exact value of the slope. You will use equations later in this chapter to obtain better values for the velocity and acceleration.

Your textbook probably shows some graphical representation of motion with graphs of position vs time, velocity vs time, and acceleration vs time. You should work through some of the examples in your textbook and ask your instructor if you have questions.

You should be able to make approximately accurate graphs of velocity and acceleration if you are given the graph of position vs. time. Some instructors do not have you make numeric calculations in creating the velocity or acceleration graphs. Important concepts regarding motion can be obtained by just estimating the value of the velocity and acceleration (especially if it is +, 0, or -) at various times. The graph is then quickly sketched.

Check your understanding of the concepts with these statements.

1. If the velocity is zero the acceleration does not have to be zero. e.g. If you toss a ball upward, at the instant the ball is at its highest position the velocity is zero but the acceleration is still -9.8 m/s² . Think this through. If you say the acceleration is zero for the ball at the top of its motion, then the velocity would not change (remember acceleration = change in velocity/time), and the position of the ball would not change (if velocity is zero the position is not changing). The ball would just hang in the air some distance above the ground!

This point is very important. Understanding the next two statements is key to understanding motion.

Velocity does not control the value of acceleration!

Acceleration controls the future value of the velocity!

2. TRUE or FALSE If the velocity is positive then the acceleration is positive.

Answer: False. The acceleration could be -, 0, or +. You must subtract two velocities to obtain acceleration. One velocity number is insufficient data to claim a certain value for the acceleration.

3. TRUE or FALSE If the acceleration is zero then the velocity is zero.

Answer: False. The velocity could be some constant value such as 5 m/s. The acceleration is found from ( 5 m/ s - 5 m/ s ) / time interval, which is 0 m/s² .

What is the meaning of an acceleration of - 3 m/s² ?

Answer: The velocity is decreasing by 3 m/s every second.

Copyright© 2017 by Greg Clements Permission is granted to reproduce this document as long as 1) this copyright notice is included, 2) no charge of any kind is made, and, 3) the use is for an educational purpose. Editing of the document to suit your own class style and purposes is allowed.Reading Guide Chapter 2 Sections 1 to 4 Page 37 in pdf file OpenStax College Physics

Chapter 1: Course Introduction, Why Study Physics, Units, Conversions, Significant Figures, Fermi Problems, YouTube Videos

Prof. Clements Notes/Hints for Physics Courses

Introduction: Chapter 1 Open Stax College Physics or Most Introductory Physics Courses

I highly recommend that you go to the web site https://openstax.org/details/books/college-physics to access the free OpenStax College Physics textbook and resources.

My notes are meant to help you understand the textbook, understand physics, and succeed in your physics course, or help you review physics as you prepare to take another course. These notes/hints are not a substitute for reading the textbook or attending lectures.

YouTube videos of my (usually short) lectures and example problems are indexed at http://www.physics.gpclements.com/ . There are also a few videos that give review of basic math tools that are used in an introductory physics course.

Terms you should know after this unit: law, theory, metric units, English units, significant figure, uncertainty

I agree with the author for the OpenStax College Physics textbook that one of the outcomes of a physics course is that the student is able to apply physics principles to a wide variety of situations. After you complete a physics course you should be better able to understand concepts and solve problems in a wide variety of science courses and careers.

Many fields of study (e.g. engineering, medical school, PT school, optometry, etc.) list physics as a required course to give students basic understanding about forces, matter, energy, properties of light, atoms, etc., and to help students develop their problem solving skills.

As soon as one experiment agrees with a hypothesis the hypothesis is not called a theory. A hypothesis gains the label of “theory” after repeatedly being tested and not found to be untrue. A theory cannot be proven true by an experiment but it can be proven to be false by an experiment. A theory becomes accepted after experiments have verified its predictions.

A theory can be given the name “law” when it has simple language and broad application.

Some would simplify the description of the scientific method to these steps:

1. Observe some process 2. Make a hypothesis 3. Perform a controlled experiment 4. Revise the hypothesis if necessary based on the outcome of the experiment, and perform more controlled experiments.

It is common to discover book titles from the 1700's and 1800's titled "Natural Philosophy." This title matches the title of “Physics” we use today.

(Optional: View https://books.google.com/?hl=en and search for natural philosophy.)

The era known as classical physics starts with Galileo's experiments (around 1600) and ends around 1900. The era known as modern physics starts around 1900. From around 1900 through the 1930s physicists became aware of the nature of the atom, the dual nature of light (particle and wave), consequences of moving at high speed, electrons, protons, neutrons, the quantum nature of the universe, the nature of the nucleus of atoms, and many more topics.

Almost 100% of physics courses use metric units (meters, kilograms, seconds, etc.). You may encounter a few problems that contain English units. You may convert the units of metric results into English units (if you live in the USA) so you can get a better understanding of the reasonableness of a result. You need to know how to convert units from English to metric and metric to English. Your textbook likely has a page in the front cover or at the end that has a table of conversion factors. You ask your instructor for help if you have trouble converting units from English units to metric units. There are also YouTube videos (see the end of this document) that will help you review this process.

The second is defined in terms of vibrations of a particular atom. The Earth's rotation is slightly variable and has a long term trend of slowing so it is no longer used to define the unit of time.

For the unit of the meter, the French surveyed a distance, determined the change in latitude (angle), and used a proportion to determine the distance from the North Pole to the equator in 1791. The distance between two marks on a platinum-iridium bar in Paris used to be “the meter.” This physical item survived World War II. But, this standard is not very portable and the distance is not accurately transferable to secondary standard bars for other countries. The meter is now defined using the equation distance = rate * time, where the rate is the speed of light and the time is a small fraction of a second (see the textbook). This definition bypasses the need for a physical object to define the distance of one meter.

The kilogram is set by a physical object that is kept in Paris. The statement in the OpenStax Physics text that an "exact replica" of the standard kilogram is kept by the NIST is perhaps a little bold. Exact measurements are not possible. The NIST kilogram is extremely close to the same mass as the standard kilogram in Paris.

You should memorize the power of ten values for these prefixes: mega, kilo, centi, milli, micro.

Two numbers called "the same order of magnitude" when they have the same “power of ten.”

You should view Table 1.3 in the OpenStax Physics text, or your own textbook, for examples of powers of 10 for distances, masses, and time intervals.

It is worth your time to perform a search on the Internet for …. Powers of Ten video .

In 2017 the Google search engine listed several sites that have this video. The video will give you some appreciation for the extremely wide range of sizes from the very small (nucleus of an atom) to the very large (large scale structures in the universe).

Conversions....I would recommend that you take the starting quantity and divide it by 1 so you have a numerator (your original number) and a denominator. Write the appropriate units on the numerator and denominator. e.g. For 80 meters/second you would have 80 meters / 1 second. This will help you to see which units are in the numerator and which are in the denominator. If you are trying to change the distance unit, the conversion factor has to have the meter unit in the denominator to cancel the meter unit in the numerator.

Converting units is a critical skill that is needed to succeed in a physics course. Review example problems in your textbook or see my YouTube videos if you need help with this.

Self Test ... Convert 1 microCentury into minutes.

Your answer should be about 52.6 minutes. This may be about the length of your lecture class in physics in the USA. So, if you think you were in class a long time, it was not really as long as a century, it was only as long as a microCentury!

Ask your instructor whether he/she makes a distinction between the terms precision and accuracy.

Uncertainty is an important concept. Scientists want to communicate how much confidence they have in a result. e.g. 6.07 cm + or - 0.02 cm represents a better known result than 6.07 + or - 0.08 cm. The first result is firmly believed to be between 6.05 and 6.09 cm. The second result is firmly believed to be between 5.99 and 6.15 cm.

Ask your instructor for guidance on handling the reporting of uncertainty for your work in the lecture part of the course and the lab part of the course. Ask your instructor for guidance on how to determine the number of “significant figures.” A digit is significant when it occurs in the placement in the number where the uncertainty starts. In the paragraph above, the 7 is significant and the number has three significant figures (6, 0, 7). The number 200 has 1 significant figure if the uncertainty is 100; it has 2 significant figures if the uncertainty is 30, and it has 3 significant figures if the uncertainty is 4. The number 0.000034 (with no uncertainty given) has two significant figures, the 3 and 4. Leading zeros for a decimal number are not significant. A basic rule is that the result of a calculation has the number of significant figures equal to the fewest number of significant figures in any data used in the calculation.

Approximation is a valuable skill. Sometimes approximation calculations are called "back of the envelope" calculations. This implies a quick calculation with little concern for precise answers. A very famous person who used this method was Enrico Fermi. He worked on the development of the atomic bomb in World War II.

Some examples of approximation can be found at http://en.wikipedia.org/wiki/Fermi_problem .

You should download the student solution manual from the OpenStax College Physics web page and work through several example problems for every chapter.

https://openstax.org/details/college-physics

Scroll down until you find the Student Resources section.

In future notes/hints you may find references to the Mechanical Universe videos. These videos are freely available online on YouTube (as of 2017)

http://www.caltech.edu/news/pioneering-physics-show-mechanical-universe-now-youtube-53331

YouTube videos of my “short” before-class lectures and example problems are indexed at

http://www.physics.gpclements.com/

Copyright© 2017 by Greg Clements Permission is granted to reproduce this document as long as 1) this copyright notice is included, 2) no charge of any kind is made, and, 3) the use is for an educational purpose. Editing of the document to suit your own class style and purposes is allowed.

This blog will cover topics for an introductory physics course.  Most of the blog posts will be comments, hints, and tips to help students understand topics taught during a full year of introductory physics classes.  The blog posts will follow the outline of topics of the OpenStax College Physics textbook.  You can get your pdf copy of this FREE textbook at https://openstax.org/details/college-physics .   I taught physics and astronomy at Dickinson College (Carlisle, PA) for four years.  I taught physics, astronomy, mathematics, and computer science at Midland University (Fremont, NE) for 32 years.  I enjoy helping others learn about physics and astronomy.  I have over 400 physics and astronomy videos posted on YouTube.  The videos include lectures and solutions for example problems.  The videos are indexed at  physics.gpclements.com and astronomy.gpclements.com (soon). Prof. Greg Clements