Fundamentals of Mechanics: Kinematics, Dynamics, and Circular Motion

Kinematics

Displacement, Velocity, and Acceleration

Kinematics is the study of motion without considering its causes. The primary quantities are displacement, velocity, and acceleration.

• Displacement (Δx): The change in position of an object. $\Delta x = x_f - x_i$

• Velocity (v): The rate of change of displacement. $v = \frac{\Delta x}{\Delta t}$

• Acceleration (a): The rate of change of velocity. $a = \frac{\Delta v}{\Delta t}$

Example: If a car moves from 0 m to 20 m in 4 seconds, its average velocity is $v = \frac{20\,\text{m}}{4\,\text{s}} = 5\,\text{m/s}$

Equations of Motion (Constant Acceleration)

For motion with constant acceleration, the following equations are used:

• $v = v_0 + at$

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

• $v^2 = v_0^2 + 2a(x - x_0)$

Example: A ball dropped from rest ($v_0 = 0$) falls under gravity ($a = -9.8\,\text{m/s}^2$).

Projectile Motion

Horizontal and Vertical Components

Projectile motion involves two-dimensional motion under gravity. The horizontal and vertical motions are independent.

• Horizontal motion: $x = v_{0x} t$ (no acceleration)

• Vertical motion: $y = v_{0y} t - \frac{1}{2} g t^2$

Example: A ball thrown at $v_0$ at angle $\theta$ has $v_{0x} = v_0 \cos \theta$, $v_{0y} = v_0 \sin \theta$.

Newton's Laws of Motion

First Law (Inertia)

An object remains at rest or in uniform motion unless acted upon by a net external force.

Second Law (Force and Acceleration)

The net force on an object is equal to the mass times its acceleration.

• $\vec{F}_{\text{net}} = m \vec{a}$

Third Law (Action-Reaction)

For every action, there is an equal and opposite reaction.

Forces and Free-Body Diagrams

Types of Forces

• Friction: Opposes motion between surfaces. Static friction: $f_s \leq \mu_s N$ Kinetic friction: $f_k = \mu_k N$

• Normal force (N): Perpendicular contact force from a surface.

• Tension: Force transmitted through a string or rope.

• Weight: $W = mg$

Example: A block on an inclined plane experiences gravity, normal force, and friction.

Circular Motion

Uniform Circular Motion

Objects moving in a circle at constant speed experience centripetal acceleration directed toward the center.

• Centripetal acceleration: $a_c = \frac{v^2}{r}$

• Centripetal force: $F_c = m \frac{v^2}{r}$

Example: A car turning in a circle of radius $r$ at speed $v$ requires a frictional force $F_c$ to stay on the path.

Work and Energy

Work

Work is done when a force causes displacement.

• $W = F d \cos \theta$

Kinetic and Potential Energy

• Kinetic energy: $K = \frac{1}{2} m v^2$

• Potential energy (gravitational): $U = mgh$

Conservation of Energy

The total mechanical energy (kinetic + potential) in a closed system remains constant if only conservative forces act.

• $E = K + U$

Table: Types of Friction

The following table compares static and kinetic friction:

Type	Equation	Description
Static Friction	$f_s \leq \mu_s N$	Prevents motion up to a maximum value
Kinetic Friction	$f_k = \mu_k N$	Opposes motion once sliding occurs

Additional info:

• Some equations and diagrams were inferred from context and standard introductory physics curriculum.

• Topics covered are typical for a first-semester college physics course focused on mechanics.