---
tags : Mechanics,Kinematics
title : 2020.4.2(4)
---
# Mechanics (4)
## §1.1 Kinematics in one-dimensional space
### Solve the differentional equation of deceleration $\frac{dv}{dt} =-\eta v$ by variable separatiion.

Do variable separation

$\large \frac{dv}{v} =- \eta dt$

Integrate it

$\large \int_{v_(0)}^{v_(t)} \frac{dv}{v} =-\int_0^t \eta dt$

This solution is

$\large logv(t) - logv(0) = -\eta t$

※$log$ is natural logarithm with base *e* ($\approx 2.71828$).

Organize them,
<font color="red">
$\huge v(t) =e^{logv(t)} = e^{logv(0)-\eta t} =v(0)e^{-\eta t}$
</font>

This equation shows that the velocity decrease gradually and reach 0 after very long time.
:::success
We call **relaxation time** that the time of velocity gets 1/$e$ times from $v(0).$
:::
Now, it is 1/$\eta$.