---
# System prepended metadata

title: '2,3D motion'
tags: [mechanics, "classic\_"]

---

# 2,3D motion
position vector $= \vec{r}$

$$\vec{r} = x\vec{i}+y\vec{j}+z\vec{k}$$

## Moving distance
$$r = \sqrt{x^2 + y^2 + z^2}$$

$$Δ\vec{r} = \vec{r_f}-\vec{r_i}$$ 

<div style="text-align: center;">
<img src="https://hackmd.io/_uploads/BkJH61xExe.png" alt="image"width="300" height="250">
</div>

:::success
$$Δ\vec{r}= (x_f-x_i)\vec{i}+(y_f-y_i)\vec{j}+(z_f-z_i)\vec{k}$$
:::
## Velocity
$$v = \frac{dr}{dt}$$

$$\vec{v}= v_x\vec{i}+v_y\vec{j}+v_z\vec{k}$$
:::success
$$vx = \frac{dx}{dt}\qquad vy = \frac{dy}{dt}\qquad vz = \frac{dz}{dt}$$
:::
## Acceleration
$$\vec{a_{avg}} = \frac{\vec{v_f}-\vec{v_i}}{t_f-t_i} = \frac{Δ\vec{v}}{Δt}$$

$$\vec{a}= a_x\vec{i}+a_y\vec{j}+a_z\vec{k}$$
:::success
$$a_x = \frac{dv_x}{dt} = \frac{d^2x}{dt^2}$$

$$a_y = \frac{dv_y}{dt} = \frac{d^2y}{dt^2}$$

$$a_z = \frac{dv_z}{dt} = \frac{d^2z}{dt^2}$$
:::
## projectile
The (a)acceleration of the projectile is constant and The acceleration direction of the projectile is parallel to the y-axis

if $a = 0$

<div style="text-align: center;">
<img src="https://hackmd.io/_uploads/SyOlAkxEgl.png" alt="image"width="300" height="250">
</div>

$$v_x = v_icos𝜃$$

$$x-x_i = (v_icos𝜃)t$$

$$v_y = v_isin𝜃$$
:::success
$$y-y_i = (v_isin𝜃)t$$
:::
if $a = g$

<div style="text-align: center;">
<img src="https://hackmd.io/_uploads/Bkdz01xVxg.png" alt="image"width="300" height="250">
</div>

$$v_x = v_icos𝜃$$

$$x-x_i = (v_icos𝜃)t$$

$$v_y = v_isin𝜃-gt$$
:::success
$$y-y_i = v_isin𝜃t+\frac{1}{2}gt^2$$
:::
* The relationship between the position, velocity and acceleration of projectile. ''without (t) variable''

* $y = y_i+Δx⋅v_itan𝜃+\frac{1}{2}g(\frac{Δx}{V_icos𝜃})^2$
$$t = \frac{Δx}{v_icos𝜃}$$

$$y = y_i+v_isin𝜃t+\frac{1}{2}gt^2$$

$$= y_i+v_isin𝜃(\frac{Δx}{v_icos𝜃})+\frac{1}{2}g(\frac{Δx}{v_icos𝜃})^2$$
:::success
$$y = y_i+Δx⋅v_itan𝜃+\frac{1}{2}g(\frac{Δx}{v_icos𝜃})^2$$
:::
* $v_y^2 = (v_isin𝜃)^2+2g(y-y_i)$

$$v^2 = v_i^2+2aS$$
:::success
$$v_y^2 = (v_isin𝜃)^2+2g(y-y_i)$$
:::
* $x_{max} = \frac{v_i^2sin2𝜃}{g}$

$$t_{max} = \frac{2v_isin𝜃}{g}$$
:::success
$$X_{max} = v_icos𝜃⋅t = v_icos𝜃⋅\frac{2v_isin𝜃}{g} = \frac{v_i^2sin2𝜃}{g}$$
:::
when $v_x = 0$
<div style="text-align: center;">
<img src="https://hackmd.io/_uploads/SJt3CyeVgg.png" alt="image"width="300" height="250">
</div>

$$v_y = gt$$

$$y = y_i+\frac12gt^2$$
## Circular motion
<div style="text-align: center;">
<img src="https://hackmd.io/_uploads/BkvV1glExg.png" alt="image"width="300" height="300">
</div>

* acceleration 

centripetal acceleration　: $a_c$

Tangential acceleration　: $a_t$

$$ a^2 = a_t^2+a_c^2$$

* centripetal acceleration  (green)

$$v = \frac{2𝜋r}{T}$$

$$a = (\frac{2𝜋r}{t})^2/r = \frac{4𝜋^2r}{t^2} = \frac{v^2}{r}$$

***
<div style="text-align: center;">
<img src="https://hackmd.io/_uploads/rkIv1xg4lg.png" alt="image"width="300" height="330">
</div>

$$\theta = \frac{S}{r} = \frac{vt}{r}$$

:::success
$$a = \frac{v^2}{r}$$
:::
* Tangential acceleration  (pink)

$$𝛼 = \frac{d𝜔}{dt}$$
:::success
$$a = r⋅𝛼$$
:::

## relatioin motion
* Velocity
if $v<<c$

$$x_{pa} = x_{pb}+x_{ab}$$

$$\frac{dx_{pa}}{dt} = \frac{dx_{pb}}{dt}+\frac{dx_{ab}}{dt}$$
:::success
$$\vec{v_{pa}} = \vec{v_{pb}}+\vec{v_{ab}}$$
:::

<div style="text-align: center;">
<img src="https://hackmd.io/_uploads/r1wtkxg4gg.png" alt="image"width="300" height="300">
</div>
