[Khan Academy] Scalars and Vectors in 1D

목차

 

Phycist's goal

- precisely define quantities in the universe

- find relationships between those quantities

So, vectors and scalars are needed.

 

Scalars

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Only magnitude exists

ex) mass, distance, speed, volume, temperature, energy

 

Vector

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Magnitude and direction exists

ex) position, displacement, velocity, acceleration

 

cf) Position and temperature

You can misunderstand position as scalar. But position is vector.

In defined coordinate system, [10m, right] and [10m, left] are different.

(Complicated explanation about position is 

 

You can also misunderstand temperature as vector. But temperature is scalar.

There's no quantity like [36.5℃, left].

 

Through these two cases, we can distinguish vectors and scalars much easier.

 

<To The Left~ method>

step 1. Put the word 'left' to certain quantity.

step 2. If it's weird, that quantity is scalar.

step 3. If it makes sense, that quantity is vector. 

 

How to expressing vector 

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Vector notation

- Draw an arrow over the quantity.

- Or write the quantity in bold text.

quantity	vector
position	$\vec x$
displacement	$\Delta\vec x$
velocity	$\vec v$
acceleration	$\vec a$

 

Expressing direction

step 1. Defined coordinate system is essentially needed.

step 2. with '+' & '-' sign

 

Drawing a vector

We can draw a vector with arrow.

- length: magnitude of a vector

- direction: direction of a vector

 

cf) Vectors can be drawn proportionally appropriately.

ex) velocity vectors [+6m/s] and [+4m/s] can be drawn with arrows which is ratio of 3:2.

 

Visually adding & subtracting vectors

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<tip-to-tail> method

step 1. Put 1st vector's tip to 2nd vector's tail.

step 2. Connect 1st vector's tail to 2nd vector's tip.

 

This method works for any number of vector in 1, 2, 3 dimentions.

 

Adding

by <tip-to-tail> method

 

Subtracting

step 1. Put '-' sign to the vector you want to subtract.

step 2. Apply <tip-to-tail> method.

 

Numerically adding & subtracting vectors

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We can do adding or subtracting vectors without drawing.

 

step 1. Define a coordinate system.

step 2. Just calculate like calculating numbers.

 

Position, velocity, and speed

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Position(vector)

length: distance in relation to an arbitary reference origin

 

Displacement(vector)

: change in position

 

$\Delta\vec x=\vec x-\vec x_0$

 

Velocity(vector)

: How quickly the position changes.

 

$\vec v=\frac{\Delta\vec x}{\Delta t}$

 

+) relative velocity

What is important in measuring velocity is 'where you are'.

 

Speed(scalar)

: How quickly the distance(: 이동거리) is travelled.

 

$speed=\frac{distance}{\Delta t}$

 

Difference of position vector and displacement vector

$\vec r_2-\vec r_1=\vec d$

 

Practice

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Practice - Scalars and vectors

[case 1]

step 1. Calculate the average velocity at time 0 ~ 10s and 10s ~ 20s.

step 2. Calculate the average of step 1's average velocities.

 

[case 2]

step 1. Calculate displacement.

step 2. Divide displacement by the amount of time.

 

[case 2] is more efficient.

 

배운 영단어

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물리적 용어

position: 위치

displacement: 변위

coordinate system: 좌표계