Calculating average velocity or speed | One-dimensional motion | Physics | Khan Academy Khan Academy Khan Academy

Example of calculating speed and velocity. Created by Sal Khan. Watch the next lesson:
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https://www.khanacademy.org/science/p... Physics on Khan Academy: Physics is the study of the
basic principles that govern the physical world around us. We'll start by looking at motion itself. Then, we'll learn about forces, momentum, energy, and other concepts in lots of different physical situations. To get the most out of physics, you'll need a solid understanding of algebra and a basic understanding of trigonometry. About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content.

Intro to vectors & scalars | One-dimensional motion | Physics | Khan Academy

Distance, displacement, speed and velocity. Difference between vectors and scalars. Created by Sal Khan.

Watch the next lesson: https://www.khanacademy.org/science/p...

Missed the previous lesson? https://www.khanacademy.org/science/p...

Physics on Khan Academy: Physics is the study of the basic principles that govern the physical world around us. We'll start by looking at motion itself. Then, we'll learn about forces, momentum, energy, and other concepts in lots of different physical situations. To get the most out of physics, you'll need a solid understanding of algebra and a basic understanding of trigonometry.

About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content.

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Basic Mathematics - Stacked Multiplication Problems

When you multiply a number or amount, you're increasing it many times. In the last lesson, you learned that multiplication can be a way to understand things that happen in real life. For instance, imagine that a store sells boxes of pears. The small boxes contain five pears each. You buy two. You could write the situation like this, and use the times table to solve it:

Now, imagine that you decide to buy two larger boxes containing 14 pears each. That situation would look like this:

This problem is harder to solve. Counting the pears would take a while. Plus, there's no 14 on the times table. Fortunately, there's a way to write the problem so that you can break it into smaller pieces. It's called stacking. It means that we'll write the numbers on top of one another instead of side by side.

Basic Mathematics - Multiplication

What is Multiplication?

When you multiply, you're increasing a number over and over again. Basically, multiplication is adding something more than once. For instance, if you eat 4 pieces of candy, then you eat another 4, then 4 more, you can say that you multiplied the amount of candy you ate.

Multiplication happens all the time in real life. For example, consider the situation below.

Imagine that you buy a 6-pack of soda. You have 1 set of 6 cans.




Writing Multiplication Expressions

As you just saw, a multiplication expression is written like this:

2 x 6

You can read that expression as two times six. The multiplication symbol (x) can also be called the times symbol. Remember, you always put it between the numbers you want to multiply.

Many real-life situations can be expressed with multiplication. For instance, imagine that you want to make three cakes. The recipe says that each cake will need two eggs. In other words, you need 3 x 2 eggs.

Basic Mathematics - Subtracting Larger Numbers

In Lesson 3, we learned that counting and using visuals can be useful for solving basic subtraction problems. For instance, say you have 9 apples and you use 6 to make a pie. To find out how many apples are left, you could represent the situation like this:

It's easy to count and see that 3 apples are left.

What if you need to solve a subtraction problem that starts with a large number? For instance, let's say instead of making an apple pie, you want to pick apples from an apple tree. The tree has 30 apples and you pick 21. We could write this as 30 - 21.

You might see why counting to solve this problem isn't a good idea. When you have a subtraction problem that starts with a large number, it could take a long time to set up the problem. Imagine the time it would take to count out 30 objects and then take away 21! Also, it would be easy to lose track as you counted. You could end up with the wrong answer.

For this reason, when people solve a subtraction problem with large numbers, they set up the problem in a way that makes it easy to solve one step at a time. Let's see how this works with another problem: 79 - 13.

In the last lesson, we learned how to write expressions. However, subtracting with larger numbers is easier when the expressions are written in a different way.

We can see that 79 - 13 and  mean the same thing — they're just written differently.

Basic Mathematics - Substraction

Subtraction is taking things away. When you have an amount and you subtract from it, the amount becomes smaller. Subtraction happens a lot in real life.


For instance, imagine we have 8 eggs...

As we saw, if you have 8 eggs and subtract 3 of them, you'll have 5 eggs left. In other words:

8 - 3 = 5

8 - 3 = 5 is a mathematical expression. You could read it like this: five minus three equals two. As we learned in Lesson 1, a mathematical expression is basically a math sentence that uses numbers and symbols. When we write a subtraction expression, we use two symbols: - and =.

The minus sign (-) means one thing is being subtracted from another. This is why we put it after the first group of eggs — we had 8 eggs and subtracted 5 of them.



The Equals Sign

The other symbol in our expression is the equals sign (=). As we learned in Lesson 1, the equals sign means two numbers or expressions are equivalent, or equal. Even though they might look different, they mean the same thing.

In our eggs example, since 3 eggs were left, we wrote 3 to the right of the equals sign. That shows each side means 3. 3 eggs on the left, and the number 3 on the right. Both sides are equal.

Basic Mathematics - Adding Larger Numbers

As we saw in Lesson 1, you can often use counting and visuals to solve basic addition problems. For instance, imagine that 3 people are going on a trip and 2 more decided to join. To find out how many people were going total, you could represent the situation like this:

Once you look at the problem visually, you can count and see that 5 people are going on the trip.

What if you have a bigger problem to solve? Imagine that a few groups of people are going somewhere together. 30 people travel on one bus, and 21 travel on another. We could write this as 30 + 21.

It might not be a good idea to solve this problem by counting. First of all, no matter how you choose to count, it would probably take a pretty long time to set up the problem. Imagine drawing 30 and 21 pencil marks on the page, or counting out that many little objects! Second, actually counting the objects could take long enough that you might even lose track.

For this reason, when people solve a large addition problem, they set up the problem in a way that makes it easier to solve one step at a time. Let's look at the problem we discussed above, 30 + 21.

In the last lesson, we learned how to write expressions. However, when we're adding larger numbers, it helps to write the expressions in a different way.

We can see that 30 + 21 and   mean the same thing. They're just written differently.

Basic Mathematics - Addition

What is addition?

Addition is a way to put things together. When you add two amounts, you're counting them together, as one larger amount. Addition happens all the time in real life.


For instance, if this rabbit has babies ...

What if there were four more rabbits?

As you can see, if you have 4 rabbits and add 4 more, you'll have 8 rabbits in total. You could write it like this:

4 + 4 = 8

4 + 4 = 8 is a mathematical expression. You could read it like this: Four plus four equals eight. A mathematical expression is basically a math sentence. It uses numbers and symbols instead of words. When we write out expressions with addition, we use two symbols: + and =.

The plus sign (+) means two things are being added together. This is why we put it between the rabbits—we had 4 rabbits and added 4 more.

The other symbol in our expression is the equals sign (=). When you see the equals sign in an expression, it means two are more things are equal, or equivalent. Things that are equivalent don't always look or seem exactly alike, but they mean the same thing.

For instance, when you see someone you know, there's a few things you might say:


These words aren't exactly alike, but they mean the same thing. They're all ways to greet someone.

In math, the equals sign shows that two numbers or expressions mean the same thing, even though they might look different. Remember our rabbits? Because there were 8 rabbits total, we wrote 8 to the right of the equals sign.

See how each side means 8? There are 8 rabbits on the left, and the number 8 on the right. Both sides are equal.

Basic Mathematics - Interests

The Compound Interest Equation

P = C (1 + r/n) nt
where
    P = future value
    C = initial deposit
    r = interest rate (expressed as a fraction: eg. 0.06)
    n = # of times per year interest is compounded
    t = number of years invested
Simplified Compound Interest Equation

When interest is only compounded once per year (n=1), the equation simplifies to:
P = C (1 + r) t
Continuous Compound Interest

When interest is compounded continually (i.e. n --> ), the compound interest equation takes the form:
P = C e rt
Demonstration of Various Compounding

The following table shows the final principal (P), after t = 1 year, of an account initially with C = $10000, at 6% interest rate, with the given compounding (n). As is shown, the method of compounding has little effect.
n P
1 (yearly) $ 10600.00
2 (semiannually) $ 10609.00
4 (quarterly) $ 10613.64
12 (monthly) $ 10616.78
52 (weekly) $ 10618.00
365 (daily) $ 10618.31
continuous $ 10618.37
Loan Balance

Situation: A person initially borrows an amount A and in return agrees to make n repayments per year, each of an amount P. While the person is repaying the loan, interest is accumulating at an annual percentage rate of r, and this interest is compounded n times a year (along with each payment). Therefore, the person must continue paying these installments of amount P until the original amount and any accumulated interest is repaid. This equation gives the amount B that the person still needs to repay after t years.
B = A (1 + r/n)NT - P (1 + r/n)NT - 1
(1 + r/n) - 1
where
B = balance after t years
A = amount borrowed
n = number of payments per year
P = amount paid per payment
r = annual percentage rate (APR)

Basic Mathematics - Introduction With Numbers

Number
Name
How many
0 zero
1 one

2 two
3 three  
4 four  
5 five    
6 six      
7 seven        
8 eight        
9 nine          
10 ten            
20 twenty two tens
30 thirty three tens
40 forty four tens
50 fifty five tens
60 sixty six tens
70 seventy seven tens
80 eighty eight tens
90 ninety nine tens


Number Name How Many
100 one hundred ten tens
1,000 one thousand ten hundreds
10,000 ten thousand ten thousands
100,000 one hundred thousand one hundred thousands
1,000,000 one million one thousand thousands
Some people use a comma to mark every 3 digits. It just keeps track of the digits and makes the numbers easier to read.

Beyond a million, the names of the numbers differ depending where you live. The places are grouped by thousands in America and France, by the millions in Great Britain and Germany.

Name American-French English-German
million 1,000,000 1,000,000
billion 1,000,000,000 (a thousand millions) 1,000,000,000,000 (a million millions)
trillion 1 with 12 zeros 1 with 18 zeros
quadrillion 1 with 15 zeros 1 with 24 zeros
quintillion 1 with 18 zeros 1 with 30 zeros
sextillion 1 with 21 zeros 1 with 36 zeros
septillion 1 with 24 zeros 1 with 42 zeros
octillion 1 with 27 zeros 1 with 48 zeros
googol
1 with 100 zeros
googolplex
1 with a googol of zeros
Fractions
Digits to the right of the decimal point represent the fractional part of the decimal number. Each place value has a value that is one tenth the value to the immediate left of it.

Number Name Fraction
.1 tenth 1/10
.01 hundredth
1/100

.001 thousandth
1/1000

.0001 ten thousandth 1/10000
.00001 hundred thousandth 1/100000
Examples:

0.234 = 234/1000 (said - point 2 3 4, or 234 thousandths, or two hundred thirty four thousandths)

4.83 = 4 83/100 (said - 4 point 8 3, or 4 and 83 hundredths)

SI Prefixes

Number Prefix Symbol
10 1 deka- da
10 2 hecto- h
10 3 kilo- k
10 6 mega- M
10 9 giga- G
10 12 tera- T
10 15 peta- P
10 18 exa- E
10 21 zeta- Z
10 24 yotta- Y
Number Prefix Symbol
10 -1 deci- d
10 -2 centi- c
10 -3 milli- m
10 -6 micro- u (greek mu)
10 -9 nano- n
10 -12 pico- p
10 -15 femto- f
10 -18 atto- a
10 -21 zepto- z
10 -24 yocto- y


Roman Numerals

I=1 (I with a bar is not used)
V=5 _
V=5,000
X=10 _
X=10,000
L=50 _
L=50,000
C=100 _
C = 100 000
D=500 _
D=500,000
M=1,000 _
M=1,000,000
Roman Numeral Calculator

Examples:

1 = I

2 = II

3 = III

4 = IV

5 = V

6 = VI

7 = VII

8 = VIII

9 = IX

10 = X
11 = XI

12 = XII

13 = XIII

14 = XIV

15 = XV

16 = XVI

17 = XVII

18 = XVIII

19 = XIX

20 = XX

21 = XXI
25 = XXV

30 = XXX

40 = XL

49 = XLIX

50 = L

51 = LI

60 = LX

70 = LXX

80 = LXXX

90 = XC

99 = XCIX

There is no zero in the roman numeral system.

The numbers are built starting from the largest number on the left, and adding smaller numbers to the right. All the numerals are then added together.

The exception is the subtracted numerals, if a numeral is before a larger numeral, you subtract the first numeral from the second. That is, IX is 10 - 1= 9.

This only works for one small numeral before one larger numeral - for example, IIX is not 8, it is not a recognized roman numeral.

There is no place value in this system - the number III is 3, not 111.

Number Base Systems
Decimal(10)
Binary(2)
Ternary(3)
Octal(8)
Hexadecimal(16)
0

0
0
0

0

1
1
1
1
1
2
10
2
2
2
3
11
10
3
3
4
100
11
4
4
5
101
12
5
5
6
110
20
6
6
7
111
21
7
7
8
1000
22
10
8
9
1001
100
11
9
10
1010
101
12
A
11
1011
102
13
B
12
1100
110
14
C
13
1101
111
15
D
14
1110
112
16
E
15
1111
120
17
F
16
10000
121
20
10
17
10001
122
21
11
18
10010
200
22
12
19
10011
201
23
13
20
10100
202
24
14
Each digit can only count up to the value of one less than the base. In hexadecimal, the letters A - F are used to represent the digits 10 - 15, so they would only use one character.