🏃 Kinematics

——— 🚗 One-Dimensional Motion ———

Time (t):
How long you were driving for. Unit: seconds (s)
Position (x):
Where you are or where you were. Unit: meters (m)
Velocity (v):
The way your position changes over time — speed with direction. Unit: meters per second (m/s)
Acceleration (a):
The way your velocity changes over time. Unit: meters per second squared (m/s²)

Average velocity:

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

(Displacement divided by time interval)

Average acceleration:

$$\bar{a} = \frac{\Delta v}{\Delta t}$$

(Change in velocity divided by time interval)

Final velocity:

$$v = v_0 + at$$

(Initial velocity plus acceleration times time)

Average velocity (constant acceleration):

$$\bar{v} = \frac{v_0 + v}{2}$$

Displacement under constant acceleration:

$$x - x_0 = v_0 t + \frac{1}{2} a t^2$$

For any physical variable that changes over time:

If it's constant, use the value directly.
If it varies, and only beginning and end are known:

$$\bar{V} = \frac{V_0 + V_{\text{end}}}{2}$$
This is an effective approximation when the exact form is unknown.

Motion in 2D is described using vectors:

$$\vec{r} = x \hat{i} + y \hat{j}$$
(In 3D, add: $ z \hat{k} $)
Velocity vectors can be decomposed into components using trigonometry:
If an object is launched at an angle ($ \theta = 30^\circ $) with speed ($ v $):
- Horizontal component:
  
  $$v_x = v \cos(\theta)$$
- Vertical component:
  
  $$v_y = v \sin(\theta)$$
Velocity vector:
$$\vec{v} = v_x \hat{i} + v_y \hat{j} = v \cos(\theta) \hat{i} + v \sin(\theta) \hat{j}$$
Apply 1D motion equations independently in x and y directions.