For more information, please see full course syllabus of High School Physics

For more information, please see full course syllabus of High School Physics

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### Math Review

- Science is almost always done in the metric (SI) system. This course will only use this system of units.
*Scientific Notation*: We can condense numbers that would require many zeros to write by using powers of 10. For example: 0.027 = 2.7·10^{−2}and 4700 = 4.7 ·10^{3}.*Significant Figures*: How many digits we have in a number tells us how much we can "trust" the number. Just because your calculator gives you a lot of digits does not mean you can trust it more than the data you started off with.*Trigonometry*: If you don't remember trigonometry, go look up a quick refresher on the basics. We won't need very complex trig in this course, but we use the core ideas a lot.*Vectors*: A vector is a way to show both length and direction. Equivalently, we can name a vector by naming its components.- When working with vectors, we add them together component-wise.
- If you multiply a scalar (a single number) with a vector, it just multiplies each component of the vector. [Notice that there is no definition for multiplying two vectors together. It wouldn't make sense!]
- We can find the length of a vector by using the Pythagorean Theorem. If →v = (v
_{x}, v_{y}), then its length is | →v | = √{v_{x}^{ 2}+ v_{y}^{ 2}}.

### Math Review

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- The Metric System 0:26
- Distance, Mass, Volume, and Time
- Scientific Notation 1:40
- Examples: 47,000,000,000 and 0.00000002
- Significant Figures 3:18
- Significant Figures Overview
- Properties of Significant Figures
- How Significant Figures Interact
- Trigonometry Review 8:57
- Pythagorean Theorem, sine, cosine, and tangent
- Inverse Trigonometric Functions 9:48
- Inverse Trigonometric Functions
- Vectors 10:44
- Vectors
- Scalars 12:10
- Scalars
- Breaking a Vector into Components 13:17
- Breaking a Vector into Components
- Length of a Vector 13:58
- Length of a Vector
- Relationship Between Length, Angle, and Coordinates

### High School Physics Online Course

### Transcription: Math Review

*Hi, welcome to educator.com. This is the beginning of Physics.*0000

*We are going to do first up, before we get into the Physics, we are going to do a quick Math review. *0004

*So, even if you feel really strong at Maths, just make sure to real quickly skin through the section, because we want to make sure you understand all these concepts clearly,*0008

*because they might come up in this course, and they might also come up in whatever course you are taking.*0015

*This is a supplement to your other Physics courses.*0020

*So, it is a good idea to make sure you have definitely got the background and the skills inside of this Math review.*0021

*Let's get started: First off, the metric system. Also, called the S.I. units, which is from the French System Internationale, which is the people who first created the metric system and first propagated its use.*0026

*So, the metric system is created in 1800's to, actually may be the sub 1800's, I should know that, anyway, I am sorry, so, anyway the metric system was created to standardize the measurements and it has done a great job at that.*0038

*Almost all the countries in the world, the exception, the United States of America are completely standardized on it, and even in the United States, everybody in Science, in Physics, they all use the metric system.*0052

*The metric system is great, and it's the way to do things. *0063

*So, the basic units we work with in Physics are, distance, just the metre denoted by a small 'm' mass, which is the kilogram, denoted by 'kg'.*0066

*I would like to point out it's the kilogram, so it's not actually the gram that we consider our basic unit of mass, we consider the kilogram our basic unit of mass, just an interesting thing to point out. *0076

*The volume, volume which comes in litres denoted by a small 'l' or sometimes a 'cursive l', if it gets confused as a '1' sometimes, and finally time, the second, which is denoted by 's'.*0085

*Scientific notation. *0101

*What if we had a problem involving the number say 47 billion or 0.00000002?*0103

*If you had to write down that number, more than two or three times, I think you would be unhappy, and I think you should be unhappy.*0111

*That is a lot of times that you have to write a number.*0117

*That many digits is just a pain, and you are not really putting in much information as you feels like in those zeros.*0120

*So we wouldn't want to write that all those times. *0124

*So, how else could we write it. The trick here is scientific notation. The idea here is, that you can convert it by using powers of ten.*0126

*So, 47000000000 is the same as 47 × 10 ^{9}, because we got 1,2,3..4,5,6..7,8,9 , so times ten to the 9th, and if we wanted to have it so we only had one digit at the very front, we could push it over for one more and we could have 4.7 times ten to the tenth.*0135

*Same idea if we want to move a digit up, we go back 1,2,3..4,5,6,7,8 spaces, so that would be 2 × 10 ^{-8}.*0154

*We are able to compact information this way because, ten to the one is equal to 10, so ten to the two is equal to hundred and so on and so forth.*0165

*We can also go negative, so ten to the negative one is 0.1, so ten to the negative two is 0.01.*0171

*This allows us to slide digits around, so we don't have to write really really long numbers, because when we are dealing with say number of atoms or the charge of an electron or the distance from here to the sun, we are going to be dealing with very large and very small numbers depending. *0177

*So Physics deals some extreme values of numbers, and we don't want to get cramps, because we are going to write 30 zeroes every time a number comes up.*0191

*Significant figures. *0199

*Also called sig figs. Significant figures are a way of showing, how precise our information is.*0200

*Since all information are susceptible to some amount of error, even if you look at a really fine ruler, it's hard tell what is the difference between one hundredth of a millimetre to the left or the right.*0206

*There is always some amount where it's a judgment call, and you might be slightly wrong.*0216

*There is always uncertainty in every measurement, there is always some little bit of possible error.*0220

*So significant figures expresses how certain we are with the measurement, or what the uncertainty in the measurement is.*0226

*It says how much we can trust our info.*0232

*Significant figures give us a way in letting us know how much we should rely on the information we have.*0236

*Which figures are significant? That requires a little thought.*0241

*Significant digit is any of the following: Any digit that is not a zero. *0246

*The zeroes between non-zero digits, and zeroes to the right of significant digits.*0250

*The only digits that are not significant are the digits to the left of significant digits, which makes a lot of sense.*0255

*If I wrote two, and if I wrote a bunch of zeroes in front of it, well, that is the exact same thing.*0260

*There is no way to measure the difference between two and a two with a bunch of zeroes in front of it.*0266

*It means the exact same thing, and there is no way you can measure the difference, and there is no significance in all of these zeroes.*0270

*We would not care about them, knock them out. *0277

*This would only have a significant digit of 'one', one significant digit, one sig fig. *0279

*Let's try some other examples: Our first one, we got 1,2,3,4.*0284

*So this one would have four significant figures.*0289

*This one has 1,2,3, what about that times 10 to the fourth. *0292

*Well, that times ten to fourth if we would have 10 to the fourth, well that would be, 1-0-3-0-0.*0297

*So, 10300. *0307

*But if we had 10300, what we would be saying is, we have precisely measured 10300. *0309

*Precisely measured 10300 metres.*0315

*But what we really did, we only managed to precisely measure the first 10300, but it might be up or down a little bit.*0317

*It could be 10349, or 10251.*0325

*It could be something that is close to what we could round, we are only sure up to that 10300.*0330

*So that is the point of the sig fig here.*0335

*So, that scientific notation also gives us the ability to show the information that we have measured for sure, but there might be some hash to just the zero. *0338

*So let us not multiply out to say the scientific notation and then find the sig figures, you find the sig figs before you multiply the scientific notation. *0347

*So this one would have three significant digits.*0355

*Here we have 1,2 and all of these are zeroes, so it just has two significant digits. *0358

*Here we have 1,2,3,4,5, so it has five significant digits because these ones don't count, but these ones do count because they are to the right*0365

*It means that you have measured something precisely.*0375

*There is a difference if I say, I weigh about 75 kilos. I weigh about 75 kilograms, or if I say I weigh 75.000 kilograms.*0377

*That means I have managed to get a really precise reading.*0388

*I am down to within a grams certainty of my weight. *0391

*So, it is a very precise reading of my weight, very precise reading of my mass.*0394

*So, that 000 at the end matters, but in the front once again there is no extra information there. *0398

*Finally, if we had 4.700, we would have 1,2,3,4.*0404

*We count from the right end, or the left in this case because there is no zeroes to the left.*0410

*So, we count zeroes on the right, here we would have four.*0415

*How do significant figures interact with one another? *0420

*If you add, subtract, multiply or divide numbers, we have to pay attention to how the significant figures interact. *0423

*The resulting numbers are only going to have as many significant figures as the lesser of the two numbers of significant figures.*0429

*The smallest number of sig figs in the number you start with becomes the number of sig figs the result has. And this makes sense.*0434

*If I know I weigh precisely, I have the mass of 75.000 kilograms but then I get on a boat with somebody else who weigh about 80 kilograms, I can't say, together we weigh precisely 80 plus 75, precisely 155.000.*0440

*I cannot do that because, I don't know, maybe they weigh 83 kilograms.*0458

*They were unsure when they told me their mass.*0462

* So, I cannot be certain of that.*0464

* It means we have to go to the least significant digits we have, which is two, which is those two digits of 80 kilograms.*0466

*We only got two significant figures. *0475

*That is the case, we wind up actually having two, round up because we have 155, it could become 160.*0479

*Here is some great examples: We have two 2 kilograms here, and 0.0803 kilograms here.*0487

*Mathematically if we add them together, we get the number 2.0803 kilograms.*0492

*But, this guy has one significant figure. *0497

*This guy has 1,2, THREE significant figures.*0500

*It does not matter, he is the smaller one, so we have to cut off after just one, and we round here, we wind up getting just two kilograms, because we only had that significant figure of 2 kilograms in the first one.*0503

*Over here, we know that we are going 6.083 metre per second so we got 1,2,3,FOUR.*0516

*Here we got 1,TWO.*0523

*So just two significant figures over here. It is the smaller one, it wins out. *0524

*So we have to round to here. This guy will manage to cause it to round up, and we will wind up going to 13 metres.*0529

*Just a quick trig review, if you do not remember your trig, that is going to really matter with time.*0539

*So brush up on that.*0541

* Pythagorean theorem, a ^{2} + b^{2}, the two smaller sides of a right triangle, equals the other side, the hypotenuse squared.*0544

*a ^{2} + b^{2} = c^{2}. *0553

*Then we have also got the trigonometric functions to relate those sides together. *0555

*The sine of θ is equal to the side opposite over the hypotenuse.*0558

*So, this would equal 'b' over 'c'.*0563

*Cosine of θ is equal to the side adjacent over the hypotenuse. *0566

*So this would be 'a' over 'c', and finally tan θ is equal to the side opposite divided by the side adjacent.*0573

*So this would be 'b' over 'a'.*0584

*Definitely important thing to remember. *0587

*Inverse trigonometric functions. What if we know what the sides of the triangle is, and we want to find the angle.*0590

*Then we use an inverse trigonometric function.*0594

*The arcsine or the sin ^{-1}, however you want to say it, because it is measuring what is the arc of that, right?*0597

*The arc that goes along with a given ratio. arcsine of sin θ equals θ.*0603

*Allows us to reverse it.*0611

*If we look this up, if we use a calculator, it gives us an answer. If we look it up in a big book, with just a look up table, it gives us an answer. *0612

*Same basic idea.*0619

*We are able to figure out all of those ratios beforehand through clever thought, and then at any time if you want to figure out what the angles are being, we just look at the book we created, look at the table, look at the calculator.*0621

*If sin θ = b/c, we could find θ with sin ^{-1}.*0631

*So θ would be sin ^{-1}(b/c), the arcsin(b/c).*0635

*We plug in numbers, and we get what the angle is.*0641

*Vectors: Vectors are a way to think about movement.*0645

* In another sense, they are a way to simultaneously consider the distance and the angle.*0647

*'v' here has gone some distance and it is up some angle.*0651

*'u' has managed to go some distance and it is up some angle.*0657

* But alternatively we could think of it as 'v' went over to the right by 4 and it went up by 5.*0660

*'u' went to the left, so it went negative two (-2), and it went up by 2.*0668

*That is the idea of a vector. We can expand this. We can do a vector addition.*0675

*If we got two vectors we can add them, we can put them head to tail, numerically you will add their components. *0679

*So, you have got 'v' and you have got 'u', v+u is just the sum of the numbers involved.*0684

*This one is 4 and -2. 4 and -2 becomes 2.*0690

*5+2 becomes 7. *0695

*There you go. As simple as that. *0701

*Subtraction is just adding by the negative version of the number. *0703

*If you want to know what the negative version of 'u' was, -u, we just put a negative sign in front of what it was originally. *0707

*We apply that in, we get (2,-2). We add 'v' to the negative version of 'u'.*0713

*2 plus, it was 4 before, we get 6.*0720

*-2 plus, it was 5 before, we get 3. Simple as that.*0725

*Scalar: Vector is a distance and a direction. *0731

*Scalars in a way, are just a number. It is a way to scale a vector.*0735

*It is a multiplication thing. You scale the vector, you change how much it grows or shrinks.*0740

*You can the length, and even flip the direction of a vector by using scalar. *0745

*You just multiply each element of a vector with it. Vectors are multi-dimensional, scalars are just one dimensional. *0750

*If s=3, then if we had 'v' as (4,5), what we have been using so far, then 3 would just be 1,...2, ...3 out.*0755

*So, 'sv' is just 3 v's stacked on top of one another.*0767

*Makes sense. v+v+v. 3 × v.*0771

*If we had -2, then we have to flip to the negative version. Here is where -v would show up. *0775

*We stack it twice, and we have got -2v.*0780

*If we want to do it numerically, we just wind up multiplying it by each component involved. *0783

*3 times (4,5) becomes (12,15), -2 times (4,5) becomes (-8,-10).*0790

*If we want to break a vector in to its components, we just do it.*0798

*We know what each of the components are, so we can see how much should we move in the 'x', how much should we move in the 'y'.*0801

*So, v _{x} and v_{y}. *0807

*If v = (4,5), then we see that the x side must be length 4, and the y side must be length 5.*0809

*Also, if we wanted to, we can say, v = (4,5), which is the same thing as (4,0) plus (0,5), which is basically what we see right here.*0817

*We added here, to here, and we get to the same spot.*0830

*If we want to find the length of a vector, we use the Pythagorean theorem.*0839

*We know what those sides are, because we know what the x-component is, we know what the y-component is.*0842

*How does the Pythagorean theorem work?*0846

*Square root of the two smaller sides, 4 ^{2} + 5^{2} equals the square of the other side.*0850

*We call it the absolute value, the magnitude, that is how we denote it.*0860

*In this case, square root of (16+25), does not wind up coming out to be a nice round number, we get the square root of 41, that is as simple as it is going to be.*0864

*But that is what its length is. If you want, you can change it into decimal using a calculator. *0879

*There is a relationship between length, angle and coordinates. *0886

*In general, if we wanted to know what arc, what it was if we knew that our vector had a length 5 and an angle 0f 36.87 degrees above the horizontal, what would be the vector, let's just make a triangle. *0889

*Our angle here is 36.87 degrees, and this here is 5, this look like a perfect time to use sin θ.*0904

*This side over here, let's call it 'y'. *0916

*So, sin θ equals y/5.*0918

*We multiply both sides by 5, so we get 5sin θ equals y.*0927

*We plug in what that θ was, 5sin(36.87).*0932

*Punch that into a calculator, multiply by 5, and we are going to wind up getting 3.*0938

*So, that side is equal to 3. Same basic idea over here. *0944

*We call this side x, and any time we are doing this, it is going to be the hypotenuse divided by the other.*0948

*Cosine equals adjacent divided by hypotenuse. *0954

*So, any time we want to know the adjacent side, it is just going to be hypotenuse times cosine of the angle. *0957

*Or if we want to know the opposite side, it is going to be hypotenuse times sine of angle. Simple as that. *0960

*'x' is going to be 5cosine(36.87),*0966

*Toss that into a calculator, and we get 4.*0975

*The vector 'v' would be its two components put together, (4,3). *0977

*If you want to check that out, 4 ^{2} + 3^{2} = 25, which is the square of 5.*0985

*Checks out by the Pythagorean theorem. We got the answer.*0991

*That is basically all the Math that we got to have under our belt if we want to get started in this Physics course. *0994

*Hope all that made sense. If it did not make sense, go back, check some of the stuff that you do not remember from trigonometry. *0999

*Just get back up to speak, because we are going to wind up using a lot of this, especially when we are talking about multi-dimensional stuff.*1003

*Alright, see you at the next...*1010

0 answers

Post by Kevin Wang on March 17 at 11:10:48 PM

Just saying, 3*(4,5) is actually (12,15), and not (12,5)

1 answer

Last reply by: Professor Selhorst-Jones

Thu Aug 18, 2016 3:34 PM

Post by Claire yang on August 18, 2016

Do vectors continue forever?

1 answer

Last reply by: Professor Selhorst-Jones

Sun Oct 11, 2015 6:36 PM

Post by Peter Ke on October 10, 2015

Hi Professor I was wondering what math background do you need for THIS course?

Because I'm still new to this lecture so I want to know the topic you need to know for math. Thx!

2 answers

Last reply by: francisco marrero

Wed Aug 19, 2015 9:58 AM

Post by francisco marrero on July 26, 2015

Hello Professor: I am forty nine years old and would like to learn physics, as I was always intimidated by it in high school and so gave up. Do you recommend that I study all of the other math courses, such as algebra and trigonometry on educator.com, in order to understand physics very well. Please let me know your advice. Thank you very much for your time.

2 answers

Last reply by: Anna Ha

Sun May 31, 2015 8:31 PM

Post by Anna Ha on May 31, 2015

Hi Professor Selhorst-Jones,

Thank you for your great lectures! I'm finding them very helpful :)

When rounding sig figs with the number 5 at the end, in chemistry I was taught to round to the nearest even, and in physics class I was taught to just round up. But considering science as a whole.. shouldn't the rule be all the same? Which one do you think is correct?

I just wanted to check when calculations are being continued, we have to bring the whole value down. not the rounded off value right?

Thank you!

1 answer

Last reply by: Professor Selhorst-Jones

Thu Apr 30, 2015 10:19 AM

Post by enya zh on April 29, 2015

In "How Significant Figures Interact" you said that we must use the sig fig of the less accurate figure for our answer.In example 2, the question was 6.083*2.1, but you rounded it to 13. Since 2.1 has 2 sig figs, shouldn't the final answer be 12.8?

1 answer

Last reply by: Professor Selhorst-Jones

Sun Oct 5, 2014 11:51 AM

Post by Zhengpei Luo on September 24, 2014

What's the essential difference between your course and the other physics course?

2 answers

Last reply by: Lexlyn Alexander

Thu Oct 23, 2014 4:36 PM

Post by adnan alsabty on July 19, 2014

Why in scalar example 3*(4,5) is equal to (12,5). Why is not (12,15) just like -2*(4,5) is equal to (-8,-10)?????

2 answers

Last reply by: Isaac Martinez

Wed Aug 31, 2016 3:49 PM

Post by justin Gwon on June 19, 2014

In the last segment of the lecture, I don't know why 5sin(36.87) is 3 ... also 5con(36.87) is 4.

1 answer

Last reply by: Professor Selhorst-Jones

Sat Jan 4, 2014 10:25 AM

Post by Karlo Wiley on January 3, 2014

In the last segment of the lecture, I'm not sure if the angle is 36.87 because I would plug in 5 * sin (36.87) and it would give me -3.686658423 and i also plugged in 5 * cos (36.87) and I got 3.377654463... It should have 3 and 4... so I did remember in the lecture there is a segment that you can find the angle if you know your sides... sin^-1 * b/c would give you the angle... so if c = 5 and b (y in the slide) = 3 then sin^-1 * 3/5 = 0.6435011088... So with this new angle i plugged in the numbers again in 5 * sin (0.6435011088) and it gave me 3 and I also plugged in 5 * cos (0.6435011088) and that gave me 4 which was your answer and checks out with pythagoreans theory... Sooo my question is am I right? honestly I still haven't taken trigonometry or physics, I'm in tenth grade taking geometry and biology.. I'm just interested in this subject so I don't know much at all about trigonometry or physics in general...

3 answers

Last reply by: Professor Selhorst-Jones

Mon Aug 26, 2013 11:23 PM

Post by Jeremy Canaday on August 16, 2013

i didn't know that tensor calculus x vector analysis/differential equations were prerequisites to high school physics.

2 answers

Last reply by: James Pelezo

Sun Mar 24, 2013 3:12 PM

Post by Al Khurasani on October 8, 2012

Shouldn't the SI unit of Volume be "Cubic Metre" ?

"In the International System of Units (SI), the standard unit of volume is the cubic metre (m3)"

[WikiPedia]

4 answers

Last reply by: Robert Mills

Thu Oct 3, 2013 12:58 PM

Post by Nigel Hessing on June 2, 2012

I don't understand why is 3 x (4, 5) = 12,5 shouldn't it be 12, 15?