An inclined plane problem solution

The fundamentals of Physics

Quadratic equation (Lesson)

LESSONS 1, 2 (8 hours of theory and practice)

Conspectus

Theoretical module 1

(according to Physics, by James S. Walker, V1,

Second Edition, Chapters 1,2,3,4 )

My name is Valentine Voroshilov. I am from Russia.

I have M.S. in Physics and Ph. D. in Education. For more than 10 years I have taught physics and mathematics to middle school, high school, college and university students. Therefore I can say with absolute confidence, I can teach any student, but only if he or she wants to learn.

I have read hundreds of lectures, but this one would be my first lecture on English.

Why do you need to learn physics? There is a very simple answer to this question.

Each person on Earth wants to be successful in his or her life. What should you know to be successful? What main skill distinguishes the successful person from the unsuccessful one?

I’ll give you the answer. There is one skill, which is crucial for success in life. That skill is problem solving skills. Every person should solve problems during his/her life. There are financial problems, professional problems, family problems, life problems. Being able to solve difficult problems means being successful. Being able to solve only easy problems means being less successful. If you want to know how to solve the difficult problems, you should pass special training. My professional experience proves that the best subject for developing problem solving skills is Physics. The most important thing that you should learn in physics lessons is how to solve physics problems. If you learn how to solve physics problems, you can solve any problem of your life.

What is necessary for to learn how to swim? It is necessary to swim. It is necessary to be in water and to work with your hands and legs. What it is necessary for to learn how to solve problems? It is necessary to solve problems. In this case, you have to work with your head, or, more exactly, you have to work with the stuff in your head.

How shall we work?

I’ll not spend a lot of time on the theory. I shall teach you how to use the theory for the solving of problems and tasks. Your main task is to understand physics. For achieving this goal you should ask me about it. You should ask a question to me. If you do not understand, what and why I do or did, you should ask me about it.

I began to write the abstracts of my lectures (you are reading it now). It is only the short text still, I almost do not make any diagrams or sketches.

OK, let's go to Physics at last.

Physics studies nature, the world, and the universe. A main question on which each physicist searches the answer is “WHY”. Why this book has fallen. Why now it is not moving? Why summer days have more light hours than winter days? Why water will turn into an ice if we place it in a freezer? Why, why, why, billions of “why”. For many of “why” we already know answers. These answers are called laws of Nature, in particular, laws of physics. Why are laws of physics necessary for us? Because we can use them for the predicting. We want to make predictions and we can make predictions using the laws. If we do it, hence at the end we shall receive that. Any prediction always consists of two parts. “If we do it” is the reason. “We shall receive that” is a consequence. For example, if we release the book - it will fall. This is a very simple prediction. Now let’s take a more difficult prediction. If we release the book from a height of 20 meters (65.62 ft), it will fall on the ground in 2 seconds. Each physicist knows how to make much more difficult predictions. Physics explains the motion of the planets and the flights of Shuttles and a nuclear explosion and how the laser works. Physics knows many laws of nature. We shall study some of them.

So, if we know laws of nature, we can make predictions.

What are predictions for? Any prediction is a tool for solving of problems! If we know how to make a prediction, we know how to solve problems! The typical problem sounds like this: “How can I achieve the necessary result”, or “How can I get the necessary effect?” Or, “What should I do to receive it”. You see? Any problem has always two parts too (same as any prediction). But we know only second part of the problem (it). Our purpose is to find the first part (what). We know a consequence, it is our result, it is that we want to receive, to get, to achieve. Our problem, the purpose, the task, the goal, the target is to find the reason which gives this result. What should we know to solve a problem? Laws of a nature! What should we be able to do to solve a problem? We should be able to use these laws, to apply these laws, to work with these laws. For this purpose, we should be able to think. Physics is the best subject to learn to think, to argue, to prove, to make deductions and decisions.

We start studying now the simplest physics laws. These laws are good for our everyday life. We look just around us and want to understand, why vehicles are moving, why leaves are falling down from trees. First, we should say, that our world is not empty. There are many things in the world. There are trees, houses, tables and chairs, airplanes, bugs, cats and dogs. Moreover, this stuff goes, drops, rolls, runs, flies. All this stuff is different, big and small, heavy and light, light and dark, warm and cold. However, there are many of issues, which are not important for physics. For example, for physics it is not important, that a book or a brick drop from a window or from a roof. Just there is something, that may drop, and it is falling. We do not need to know the correct name of that stuff, we just call (name, describe, mark) it with such words as a body, an object, an item. Very often we use just “a particle” for any small object.

For physics it is not important also from where the body drops, either from a window, from a roof, or from an air balloon. Important only from what height the body drops. Each body, object, item has a set of different parameters, properties, characteristics. For example, any apple has a size, a weight, a color, a taste, an odor, a temperature. However, it is not important for us. Also for us it is not important from what tree drops the apple, what kind it was, whether there is a worm in it or not. If we want to find out, what time the apple will be falling, it is important to us just to know from what height it drops.

So, there are fewer things, which are important for physics, in comparison to the literature or sociology. Therefore, physics can be learned much easier than other subjects can.

What things are important for physics? First, those things are important for physics, which can be measured.

Physics begins with the measuring. What can we measure? A length, a width, a height, a distance, an area, a volume, a velocity, a weight, a mass, a temperature, a power etc. These are the most widespread physical quantities.

What does it mean to measure? For example, what do we do to measure length of the table? We take a ruler, we put it along the table and we look at the numbers. We have an object, which we measure - the table, and there is an object with which we measure - the ruler. We know the length of the ruler (or of any parts of it). We see the numbers on the ruler. And now we compare the length of the table and the length of the ruler. That’s it! The length of the table is 4.45 ft.

Any measuring is constructed in the same way. There is something, what we want to measure - object of measuring. There is with what we measure - a measuring instrument, a measure device, a tool for measuring. This measuring instrument is called often the etalon, or a sample. A ruler is a sample for any length.

At last, there is a procedure of measuring. At the end of all this procedure is a comparison. We compare an object and an etalon. We compare the table and the ruler. Even more, we want to find out what part of the ruler is exactly enough for covering the table.

Physics has some the base quantities. These are the quantities for which special etalons is manufactured. There is an etalon for length, there is an etalon for mass and there is an etalon for time. There is not an etalon for a velocity. We can make it. However, it is not necessary for us, because we can measure velocity using etalons of length and time. There is no etalon for a force. Because we can measure a force using length, mass and time etalons.

Nature does not have etalons; Nature does not need any etalons. People have invented etalons. It is possible to invent many different etalons for measuring a length. For example, in German one meter is used as the etalon of length. But in America the etalon of length is 1 ft. When we tell, that the length of a table is 4.45, this information is useless for us. We should say 4.45 what the length is, either 4.45 meters, or 4.45 yd or 4.45 ft. Therefore physicists have invented special labels, special marks for each physical quantity. This label is called “a unit” and we have to put it after the number, which means a magnitude of the physical quantity. For example, we must write, that the length of the table = 4.45 ft.

Physicists are lazy people; they do not like to do too much. Therefore, instead of words “the length of a table” they write some character too, for example L. Then the statement “the length of a table is 4.45 ft”, can be written shortly L = 4.45 ft. Here L represents the name of a physical quantity, which we measure. 4.45 is the magnitude of the quantity, ft is the unit. A unit means what kind of an etalon was used for measuring. We have to know units for main physical quantities; therefore, we should learn this table. This table contains units for both the British system of units and International System of units (shortly, SI; this system is called also the metric system).

*1) the Table

All these physical quantities are used in various laws of physics, which we shall study. Each law of physics is the formula where there are various quantities. We may multiply and divide, add and subtract these quantities. It means that we should multiply and divide, add and subtract the magnitudes. We should use mathematical rules for this purpose. But what we should do with units?

First, we have to understand that any unit is more than just a label. Unit is an algebraic symbol, which is multiplied on the magnitude of a physical quantity. L = 4.45 ft means that the length of the table is a product of the number 4.45 and the etalon 1 ft. There was 1 ft taken 4.45 times for getting the table of this length!

Second, we can use any laws from algebra to manage units. First of all, we may divide and multiply units as well as numbers. But there are two important rules, which we are obligated to execute. 1. We should use units only from one system of units. It is impossible to mix the British and International systems of units. Either the British only or the SI only. If our problem has a mix of units, we should transfer a part of units from one system to another.

Any law of physics is a formula, an equation, which contains various physical quantities. We may multiply and divide, add and subtract these quantities. Hence, we should do the same algebraic operations to units. After this manipulation we have a unit of the entire formula (“a total unit”), which is called often a dimension.

The rule # 2. For any formula, the dimensions of the two sides of the formula must be the same. In other words, the total unit of a right side of the equation should be exactly the same as the total unit of a left side of the equation. For example, it is never possible to have such formula as 2 m = 5 m because 2 is not equal 5. Also it is never possible to have such a formula as 2 m = 2 s. Because for both SI system and the British system length may never be expressed in seconds, and time may never be measured in meters.

*2) Well, let's get some practice (Problems on conversion of units, on finding a total unit, on definition of the possibility for existence of a formula)

Let's go further.

So, we can measure physical quantities, and we can calculate physical quantities. It is very important to understand, that any measuring always has an error, inexactness, inaccuracy. Every measurement has some degree of uncertainty. We never can measure something precisely. For example, we measure a length of a table. We are sure, that digit 4 is the precise digit, following digit is precise too, but the last digit can be a little more or less. We are not sure about this last digit. This digit is called a doubtful digit, we doubt about what magnitude this digit has.

Always, when we do measure something, there are digits, which we are sure about. However, at least the last digit has an uncertainty always. The more exact the measuring is, the more digits we may write with certainty. These digits are called valid digits. Valid digits together with doubtful digits are called significant figures. The more exact measuring is, the more significant figures we have.

*3) Examples for various amount of significant figures.

If always we have errors of measuring, hence it is not necessary for us to do precise calculations. For example, we measured the length L, the height H and the width W of a box.

L = 1.21 m, W = 0.85 m, H = 0.76 m. If we calculate the volume of the box we receive V = L*W*H = 1.21*0.85*0.76 = 0.78166 m³

But we cannot measure a volume more precisely than length, width and height. Therefore, we may trust only digits 0 and 7. The next digit could be a little more or less than 8. The digits 1, 6 and other 6 don’t make any sense. So we should record result as V = 0.78 m³ (Read the textbook about “round up” and round-off errors).

If we do not require precise calculations, we may not use precise formulas. We may use not precise formulas. We may use approximate formulas. We may use an approximation.

Let's receive one of approximate formulas. (1 + x)² ≈ 1 + 2*x

First we have to write the precise formula (1 + x)² = 1 + 2*x + x².

If the magnitude of x is very small, for example x = 0.01, hence the item 2*x will be equal 0.02, but the item x² will be equal 0.0001. This number is 200 times less than 0.02; therefore, we may just forget about it, i.e. just remove it from the formula. Then we shall receive (1 + x)² ≈ 1 + 2*x!

For example, we want to find the area of a quadrate (square). We measured its length and width L = W = 1.06 m. The area is A = L*W = 1.06*1.06 = 1.06² = 1.1236 m².We already know, that last two digits have no sense, i.e. actually A = 1.12 m². Exactly this magnitude we will receive if we use the approximate formula instead of the correct formula. A = 1.06² = (1 + 0.06) ² ≈ 1 + 2*0.06 = 1 + 0.12 = 1.12 m².

The approximate formula gave to us the same result, as the precise formula!

Sometimes it is much easier for us to use an approximate formula than precise one.

Let's record some approximate formulas which can be necessary for us.

Everywhere, in each formula x is any very small number.

( 1 + x)ⁿ ≈ 1 + n*x

We may take different numbers n and gain different approximate formulas.

In particular for n = 2 we have again (1 + x)² ≈ 1 + 2*x

n = 3 (1 + x)³ ≈ 1 + 3*x

n = 1/2 (1 + x)^1/2 = √(1 + x) ≈ 1 + (1/2) x = 1 + x/2

n = -1 (1 + x)^-1 = 1/(1+x) ≈1 + (-1) x = 1 - x

n = -2 (1 + x)^-2 = 1/(1+x)² ≈ 1 + (-2) *x = 1-2*x

n = -1/2 (1 + x)^-1/2 = 1/√(1 + x) ≈ 1 + (-1/2) *x = 1 - x/2

In addition, there are more formulas from trigonometry and algebra.

sin (x) ≈ x

cos (x) ≈ 1 - x²/2

e^x = exp(x) ≈ 1 + x, here e ≈ 2.72

Let's test some of the formulas with the help of the calculator.

What kind of a test we should make? Take small number x, calculate correct the magnitude of the left side of the formula, and then calculate the approximate magnitude (the right side), and at last compare the results.

Let x = 0.01 Here x has three significant figures. Then 1 + x = 1.01. We take the formula (1 + x)³ ≈ 1 + 3*x. The left side give to us 1.01³ = 1.0303 ≈ 1.03 (We have left just three significant figures again) The right side give 1 + 3*0.01 = 1.03 The same value!

Check the remaining formulas by yourself.

So, any physical quantity can be measured, any physical quantity can be calculated.

Any physical quantity has a magnitude. Almost any physical quantity has a unit.

However, there are physical quantities, which have a direction also. These quantities are guided in some direction. For example, it is a velocity. If the body has a velocity, hence the body is moving. But any motion always is guided somewhere. The vehicle may go to the north or to the east; the stone can fall downwards or go upwards etc. Also, a force always is guided somewhere. I push the book; I push it to the left! Hence, I apply the force to the book, and this force is directed to the left. The force has the direction!

However, the mass of this book is same for any directions of moving. Mass has not any direction.

So, there are physical quantities which are not guided, which do not have any directions. Such quantities are called a scalar. Mass is a scalar. There are physical quantities, which have a direction. Such quantities are called a vector. Velocity is a vector. Force is a vector. We should distinguish a scalar and a vector; therefore, we should invent different labels (marks) for them. Ordinarily we denote a scalar by just a letter (a character). We denote a vector often by a bold letter or by a letter with an arrow above it. In a class, I shall draw an arrow above a letter for any vector. But on these pages I use bold letters. On a drawing, on a diagram, or on a figure we will represent a vector as an arrow. The length of the arrow means a magnitude of a physical quantity, and the direction of the arrow means a direction of a physical quantity. For example, if a plane flies to the north we shall draw its velocity as such an arrow.

*4) We know, that algebra has special laws for numbers, these are laws of addition, multiplication and others. I shall write main of them (the laws of algebra). There are special laws for vectors. We can multiply a vector on a number, we can add a vector to another one (the rules for activities with vectors)

We already talked that there are many things in the world. These things are moving or moved, but first of all they are located somewhere. Each thing, each object, each body has a location on the ground, under the ground, in the air or at the ocean, in a space etc. Each thing, each body has the place, has the position. If we want to find out a position of an object, we have to ask “where”. Where is my vehicle now? Where is Osama bin Laden now? Physics has a special universal way to reply on a question “where”.

Let’s take a look for some body. It is doesn’t matter for us what color or size the body is. We ignore this attributes of the one now. We imagine any object as just a small particle with no size, i.e. as a dot. This dot has a position in the space.

First, we know that any object can be moved in three directions. We say usually, that the body can be moved forward or back, to the left or to the right, upwards or downwards. Hence, our particle too. We can put a ruler along each the direction and measure the position of the particle. Therefore, we should know three numbers to spot a position of the particle. There is a special procedure to find these three numbers. There are many such procedures. However, only one is important for us.

First, we select an origin. We may make any choice for it. We can take any convenient point of space as origin. Usually this point (i.e. origin) is bounded with some body from which we want to start to do the measuring (the Earth, a car etc.) Then we put a ruler # 1. This ruler is called axis X. Then we put a ruler # 2. This ruler is called an axis Y. At last, we put a ruler # 3. It is an axis Z. Now for any particle we can spot its position precisely. For this purpose, we should lead very accurately three lines from a particle up to each of axes. Thus the line which we draw from a body up to an axis should be parallel a plane which is made with two remaining axes. Points on axes, which these lines will cut, are called coordinates of a particle (or, simply, coordinates of a body). We shall consider the simplest version. Let's assume that between any of axes the angle is 90 degrees. Between the axis X and the axis Y is a right angle, between the axis Y and the axis Z is a right angle and between the axis X and the axis Z is a right angle too. To find coordinates of a particle we just should conduct a perpendicular from a particle up to each of axes. The origin has coordinates (0,0,0). Any different particle has coordinates (x,y,z). We can imagine that all the space is filled with a grid of strait lines. It would look like a three-dimensional “graph paper”. Three lines are going through any point of the space, i.e. through any particle, which there is in the space. It is easy to see on the figure how the coordinates of the particle are found. This kind of coordinates, which uses a rectangular grid, is called rectangular coordinates.

*5) Let's get some training (figure, finding of coordinates of a body).

There is a special vector, which is used in physics very often. This vector begins in an origin and finishes in that place where there is a particle. If a particle goes, this vector goes after it. Such vector is called a position vector (or radius-vector of a particle). We denote it by R. We may draw a position vector for each particle. Each position vector may be the sum of three special vectors. We may see it on a figure. We draw vector, then another, then the third. We have begun from an origin and have finished there, where there is the particle. The first vector goes along axis X. It is called the component X of vector R and denoted by Rx. The second vector goes in parallel axis Y, it is called the component Y of vector R (Ry). The third vector goes in parallel axis Z, it is a component Z of vector R (Rz). By a rule of vector addition R = Rx + Ry + Rz.

It means that if we know components of a position vector, hence we can find the vector.

On the contrary, if we know a position vector, we easily can find its components. For this purpose, we should conduct perpendiculars from the end of the position vector to an axis and conduct vector for each component.

*6) Examples, the figure.

There is a more common object then a component, it is a projection of a vector to this axis. We may draw any axis, we may draw any vector and then we may draw a projection of this vector to this axis. In the future, we shall draw very often vectors and its projections.

*7) Let’s see more details at the figure. It is a right angle and it is too. It is a vector. It is its projection. We conduct a segment. The length of this segment is equal to length of a projection. The vector, the segment and this segment form a triangle. This triangle has a right angle; this angle is 90 degrees. It is a rectangular triangle. Let's recollect some geometry. This leg is a hypotenuse, a hypotenuse always opposite to a right angle. This leg is a cathetus and this leg the cathetus too. Let's mark the legs and angles. Now we can write some formulas, which will be necessary for us.

*8) Formulas of a rectangular triangle (Pythagorean Theorem, sinus of an opposite angle, cosine of an angle, a tangent of an opposite angle)

*9) Let's consider some examples.

OK, let's go further.

Any law of physics grows up from observations. We look at the nature, we are surprised, we ask about, we think about, we observe bodies, events and facts, and we observe processes and phenomena. We invent words for describing of these bodies, processes, facts. We describe these phenomena. Writers describe these phenomena too; journalists describe these phenomena too. However, they use the other language; they use the literary language. We use the special language; we use the physics language.

Let's look at this particle. It goes in a space. In different moments, the particle takes different places, different positions, different points. If we take a pencil and mark all these positions we‘ll receive a line along which the particle was sliding. Sometimes we may see such line in the sky. The military plane draws such a line in the sky. I can take chalk and go with it along a blackboard. Hence, we shall see a line on a board along which the chalk was sliding. We have the special name for such a line. The line along which a particle goes is called the trajectory of the particle. The trajectory may have the many different forms (shapes). For example, a vehicle goes on a highway. The trajectory of the vehicle is a straight line. The satellite rotates around of the Earth. The trajectory of a satellite is a circle. Paul Pierce has thrown a ball to a basket. The trajectory of the ball is a parabola.

Not only the form of a trajectory is important for us. We want to know how many miles traveled a particle along its trajectory, how long and how fast it was moving. The length of a trajectory is called a distance. A distance is total length of travel. How can we measure a distance? There is one universal way to do this. It is necessary to draw a trajectory, then to take the long rope, to put this rope accurately along the trajectory. Then we very accurately have to remove the rope from a trajectory, to make it straight and to put along a ruler. We shall see the length of the part of our rope on the ruler. This is a length of the trajectory too. This is the distance! Certainly, there are many other ways to find a distance. We may apply laws and formulas, which, there are in geometry. The trajectory may have the shape of either a circle, or a triangle, or a quadrate etc. If it necessary, we shall write formulas for each such a figure later.

Time is a necessary factor for any motion. When the particle begins to move, we can turn on a stopwatch. When the particle finishes the moving we turn off a stopwatch and we look, how much time is passed (elapsed) from the moment of the beginning of a motion of the particle to the moment of the ending of a motion. The initial moment of time is a moment when the stopwatch was turned on. The final moment of time is a moment when the stopwatch was turned off. The time which has passed from the beginning of a motion to the ending is called an elapsed time (or time taken, or a moving time interval).

So, we need coordinates to know a position of a particle. We need a clock to know a time of event. A coordinate grid together with clock (stopwatch) is called a reference frame.

We must use a reference frame to solve any physics problem or task.

We always have an initial position of a particle and a final position of a particle. We always have an initial coordinates of a particle and final coordinates of a particle. It is not important, whether we know these coordinates or not, whether we can calculate them or not. That is important, that these coordinates are always there! We always may draw a vector from an initial point to a final. Such vector is called a displacement vector. Displacement is change in position of a particle.

*10) Look. We have a position vector for an initial point (Ri). We have position vector for a final point Rf. We have a displacement vector (S). These three vectors are always forming a triangle! By a rule of vector addition, Ri + S = Rf.

So, once again, let’s repeat the names of physical quantities and categories, which we should know: an axis, a reference frame, an origin, coordinates, a position vector, a trajectory, a distance, a displacement vector (shortly just displacement), an initial moment (a stop watch was turned on), a final moment (a stop watch was turned off), an initial position of a particle, a final position of a particle, an elapsed time.

Let's make a following step. We want to know how fast a particle was moving from an initial position up to a final one. For this purpose, we invent a new physical quantity – velocity. More precisely to say - some velocities. On definition, the ratio S/(t_f - t_i) is called an average velocity. Here S is a displacement vector, t_i is an initial moment of movement, t_f is a final moment of movement. (t_f - t_i) is exactly elapsed time of a moving. We denote an average velocity by Vave. So Vave = S/(t_f - t_i).

We have divided the vector on a number, hence we had got a vector again! An average velocity has a direction and it always points the same as a displacement vector.

The ratio of a distance to a time of a motion is called an average speed. Vave = D/(t_f – t_i). It is always a scalar (The letter V is not bold!).

The particle may be moved faster or slower. For example, the train speeds up when it goes from a T-station. Hence, right at the beginning of a motion the train goes more slowly than in the middle of its moving. Often we want to know not only an average velocity, but also velocity at each a moment! We can make it easily. If we want to find a velocity of a particle at the moment t we just should calculate an average velocity of this particle for an elapsed time from t up to t', but the moment t' should be very very very close to the moment t. t' should be so close to t , that we almost might not distinguish (separate) them each from other. So, the quantity V = S_t/(t - t') is called instantaneous velocity at the moment t, when t' comes very close to t. If we look at a figure closely we shall see, that instantaneous velocity is always points tangentially to a trajectory.

Instantaneous speed V is a magnitude of instantaneous velocity.

So, the particle may have a different instantaneous velocity for different moments. Velocity of a particle is changeable; velocity may increase and may decrease. An initial velocity V_i is velocity, which a particle has at an initial moment. A final velocity V_f is velocity, which a particle has at a final moment. An average acceleration Aave is quantity which is equal Aave = (V_f - V_i)/(t_f - t_i). An average acceleration is a vector. The instantaneous acceleration at a moment t A_t is quantity which is equal A_t = (V_t - V_t')/ (t – t'), here V_t is an instantaneous velocity at a moment t, V_t' is an instantaneous velocity at a moment t', and t' aims to t.

If instantaneous velocity of a particle is changing (whether by the magnitude or by the direction, or both), hence the particle always has an instantaneous acceleration.

You should read about all these quantities in the textbook; the textbook has more a detailed information. I want to give you more time for a problem solving. Let's get some training.

*11) Problems on a trajectory, a displacement, a distance, velocity and acceleration.