Mechanics: Kinematics and Dynamics

TL;DR

Kinematics describes how things move (position, velocity, acceleration), while dynamics explains why they move (forces). Newton's Laws are the foundation for understanding how forces cause these changes in motion. By combining these ideas, you can predict and explain the motion of almost anything.

1. The Mental Model

Think of motion in two parts: first, describing the journey itself, and second, understanding what pushed or pulled to make it happen. It's like describing a car trip versus explaining why the car sped up or slowed down.

2. The Core Material

Kinematics: Describing Motion

Kinematics is all about the "how" of motion. We use a few key terms to describe it:

• Position ($\vec{x}$ or $\vec{r}$): Where an object is. It's usually measured from an origin (a starting point). For 1D motion, it's just a number; for 2D or 3D, it needs coordinates.
• Displacement ($\Delta\vec{x}$ or $\Delta\vec{r}$): The change in position. It's the straight-line distance and direction from the start to the end point, not the total path traveled.
• Distance: The total length of the path traveled, regardless of direction.
• Velocity ($\vec{v}$): How fast an object's position changes and in what direction. It's displacement over time ($\Delta\vec{x} / \Delta t$). Speed is just the magnitude of velocity (how fast, ignoring direction).
• Acceleration ($\vec{a}$): How fast an object's velocity changes. It's change in velocity over time ($\Delta\vec{v} / \Delta t$). Acceleration means speeding up, slowing down, or changing direction.

For constant acceleration, these equations are super handy:

1. $v = v_0 + at$
2. $\Delta x = v_0 t + \frac{1}{2}at^2$
3. $v^2 = v_0^2 + 2a\Delta x$
4. $\Delta x = \frac{v_0 + v}{2}t$

Here, $v_0$ is initial velocity, $v$ is final velocity, $a$ is constant acceleration, $t$ is time, and $\Delta x$ is displacement. Choose the equation that best suits the information you have and what you need to find.

Dynamics: Explaining Motion

Dynamics addresses the "why" of motion – forces! Sir Isaac Newton's three laws are the pillars here.

• Newton's First Law (Inertia): An object at rest stays at rest, and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force. This means objects resist changes to their current state of motion.
• **Newton's Second Law ($\vec{F}_{\text{net}} = m