---
tags : Mechanics, Kinematics
title : 2020.3.31(1)
---
# Mechanics (1)
## §1.1 Kinematics in one-dimensional space

### 1.1.4 position
We can write the position *x*(*$\Delta$t*) which moves after very short time $\Delta$*t* , if we know the position *x*(0) and the velocity *v*(0) as

#### *x*($\Delta$*t*) $\approx$ *x*(0) + *v*(0) $\Delta$*t*

※"**A $\approx$ B**" means "**A is near by B from the view of number.**"

Above relation is lead by changing the average velocity to *v*(0) , so it consists accurately in small extreme $\Delta$*t.*

**(ex)** set the start position at *x*($\Delta$*t*) , and write the position *x*(2$\Delta$t) when the time 2$\Delta$*t*
#### *x*(2$\Delta$*t*) $\approx$ *x*($\Delta$*t*) + *v*($\Delta$*t*)$\Delta$*t*

Combine above 2 equation,
#### *x*(2$\Delta$*t*) $\approx$ *x*(0) + (*v*(0) + *x*($\Delta$*t*))$\Delta$*t*
Repeat n times,
#### *x*(n$\Delta$*t*) $\Delta$ *x*(0) + $\sum_{i=0}^{n-1}$*v*(*i*$\Delta$*t*)$\Delta$*t*

Therefore, we can figure out the position *x*(*t*) in time *t* by additioning past velocity to the start position *x*(0).
In addition ,we can also write acculatery by the integration defined by extreme of infinitely small section $\Delta$*t* $\to$ 0 and infinite number of divisions *n* $\to$ $\infty$
<font color="red">
$\huge
x(t) = x(0) + \int_0^t v(t') dt'$