High School Physics 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 × 10⁹, 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 right0365

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, a² + b², the two smaller sides of a right triangle, equals the other side, the hypotenuse squared.0544

a² + b² = c².0553

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, 4² + 5² 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, 4² + 3² = 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