---
# System prepended metadata

title: Motion of Particle
tags: [Mechanics, ' ss', ' ncu']

---

---
tags: Mechanics, ss, ncu
author: N0-Ball
title: Motion of Particle
GA: UA-208228992-1
---

# Motion of Particle

[Toc]

## Kinematic Equations

$$
\frac{d \vec x}{dt} = \vec v
$$

$$
\frac{d \vec v}{dt} = \vec a
$$

**Newton's Second Law**
$$
\vec F = m \vec a = m \frac{d\vec v}{dt}
$$

## Linear Momentum & Momentum Therorem

Linear Momentum $\vec p$
:    $\vec p = m \cdot \vec v$

Since force $\vec F = m \cdot \vec a = m \cdot \frac{d \vec v}{dt}$

### The differential momentum therorem

- For one dimensional problem:
$$
\frac{dp(t)}{dt} = \dot p(t) = \vec F \\[1em]
\frac{dp(t)}{dt} = \dot p(t) = m \cdot \dot v(t) = F[x(t), v(t), t]
$$
- Fow two dimensional problem:
$$
\frac{d \vec p}{dt} = \vec F
$$

:::warning
Relatistic momentum
:::
$$
\vec p = \gamma m \vec v
$$
where
$$
\gamma = \frac{1}{sqrt(1 - (\frac{v}{c})^2)}
$$

## The Integral Momentum Therorem 

$$
\vec p = \int\vec F dt
$$

**or**
$$
p(t_2) - p(t_1) = \int^{t_2}_{t_1} F(t) dt
$$
$$
p(t_2) - p(t_1) = \int^{t_2}_{t_1} F[x(t), v(t), t] dt
$$

# Kinetic Energy & Energy Theorem

$$
T = \frac{1}{2} m v^2 = \frac{1}{2} m \vec v \cdot \vec v
$$

## Divergence form

- For one dimensional form
$$
\frac{dT(t)}{dt} = mv \frac{dv}{dt} = v F
$$
- For two up dimensional form
$$
\frac{dT}{dt} = m \vec v \cdot \frac{d \vec v}{dt} = \vec v \cdot \vec F = Power
$$

## Integrated form 

- One dimensional form
$$
\Delta T = \int^{t_2}_{t_1}F(t)v(t)dt
$$
- Two or three dimensional form
$$
\Delta T = \int^{t_2}_{t_1}\vec F[\vec x (t)] d \vec x
$$

## One dimentional motion of a partical

$$
\frac{dv(t)}{dt} = \frac{F}{m} \Rightarrow v(t_1) = v(t_0) + \int^{t_2}_{t_1} \frac{F(t)}{m} dt
$$

## Damping force

$$
\begin{aligned}
\frac{dv}{dt} &= \frac{F(v)}{m} = \frac{1}{m}(-bv) \\[1em]
\frac{1}{v} dv &= -\frac{b}{m} dt \\[1em]
\Rightarrow ln\frac{v_1}{v_0} &= - \frac{bt}{m} \\[1em]
\Rightarrow v_1 &= v_0 e ^{\frac{-bt}{m}}
\end{aligned}
$$

## Conservation Force

$$
\begin{aligned}
&\frac{dv}{dt} = \frac{F(x)}{m} = \frac{1}{m}[-\frac{dV(x)}{dx}] \\[1em]
&\Rightarrow v dv = \frac{1}{m}[-\frac{dV(x)}{dx}] v dt =  \frac{1}{m}[-\frac{dV(x)}{dx}] \frac{dx}{dt} dt \\[1em]
&\Rightarrow \int v dv = \int \frac{-1}{m}dV(x)\\[1em]
&\Rightarrow \frac{1}{2} (\Delta v)^2 = \frac{-1}{m} \Delta V + C\\[1em]
&\Rightarrow \frac{1}{2} m (\Delta v)^2 + \Delta V = C
\end{aligned}
$$

## Falling bodies

$$
\begin{split}
& \quad \frac{d^2 z}{dt^2} = \frac{dv_z}{dt} = - \frac{mg}{m} = -g \\[1em]
&\Rightarrow \int d \left(\frac{dz}{dt}\right) = \int -g dt \\[1em]
&\Rightarrow \int dz = \int -gt dt \\[1em]
&\Rightarrow z = - \frac{g}{2} t^2
\end{split}
$$

## Simple Harmonic oscillator

$$
m \ddot x = -kx \Rightarrow \ddot x = - \frac{k}{m}x
$$

$$
x = \cos \left(\sqrt{\frac{k}{m}} t \right), \quad \omega ^2 = - \frac{k}{m}
$$

## Linear differential equations with constant coefficients

$$
a_2 \ddot x + a_1 \dot x + a_0 x = f(t)
$$

**Laplace transfomation**

## Damped Harmonic Oscillator

$$
m \ddot x + b \dot x + kx = 0
$$

**Let $x = e^{\lambda t}, \quad \dot x = \lambda e^{\lambda t}, \quad \ddot x = \lambda^2 e^{\lambda t}$**

$$
\begin{split}
& \quad \left (\lambda^2 + \frac{b}{m} \lambda + \frac{k}{m} \right) e^{\lambda t} = 0 \\[1em]
& \Rightarrow \lambda = \frac{-\frac{b}{m} \pm \sqrt{\left(\frac{b}{m}\right)^2 - 4 \frac{k}{m}}}{2}
\end{split}
$$

**Let $\gamma = \frac{b}{2m}, \quad \omega_0 = \sqrt{\frac{k}{m}}$**

### Case 1 $\gamma^2 \gt \omega_0^2$
$$
\begin{split}
& \quad \lambda_{
\begin{array}{c}
1 \\
2
\end{array}
} &= - \gamma \pm \sqrt{\gamma ^ 2 - \omega_0^2} \\[1em]
&\Rightarrow x &= c_1 e ^ {\lambda_1 t} + c_2 e^{\lambda_2t}
\end{split}
$$

### Case 2 $\gamma^2 = \omega_0^2$

$$
\begin{split}
&\quad \lambda = - \gamma \\[1em]
&\Rightarrow x = (c_1 + c_2 t) e ^{-\gamma t}
\end{split}
$$

### Case 3 $\gamma^2 \lt \omega_0^2$

$$
\begin{split}
& \quad \lambda_{
\begin{array}{c}
1 \\
2
\end{array}
} &= - \gamma \pm \omega i, \quad \omega = \omega_0 \sqrt{1-\left( \frac{\gamma}{\omega_0}\right)^2} \\[1em]
&\Rightarrow x &= e^{- \gamma t} \left[A \cos \left( \omega t\right) + B \sin \left( \omega t \right) \right]
\end{split}
$$