Kinematics

Preface

Introduction

Kinematics is the branch of physics dealing with the motion of particles or bodies. It defines movement at the level of position, velocity and acceleration, without incorporating masses and forces.

Prerequisites

A proper study of the subject requires an understanding of 2D and 3D vectors. Of course you also need to review basic algebra, trigonometry and calculus. A proper understanding of kinematics require an understanding of vector calculus too.

Sample Problems

Chapter 0: A Mathematics Review

1. Algebra Review
2. Geometry and Trig Review
3. Calculus Review
4. Vector Review

Chapter 1: Plane Kinematics

1. 1D Coordinate Systems
2. 2D Coordinate Systems
3. Linear Motion
4. Transformations

Chapter 2: Kinematics in Three Space

1. 3D Coordinate Systems
2. Particles

Two Dimensional Kinematics

Displacement, Velocity, and Acceleration

Displacement is the change in position with respect to the frame of reference. In the case of a car moving on a road, the displacement is the change in position with respect to the road. This is important because the Earth itself is rotating and orbiting at considerable speed, which you do not notice due to inertia. Since omnipotence is not possible, all observations are made from a specific frame of reference. Displacement is usually expressed as a length measurement. Velocity is the change in displacement with respect to a change in time. The velocity of an object is again relative to the frame of reference. Acceleration is the change in velocity with respect to a change in time. Therefore, we know that, for functions for displacement, velocity, and acceleration, respectively d, v, t :

$d(t)=\int _{0}^{t}v(x)dx$

$v(t)={\frac {dd}{dt}}=\int _{0}^{t}a(x)dx$

$a(t)={\frac {dv}{dt}}$

Kinematics Formulas

Using the above can create the basic kinematics formulas(subject to acceleration being constant):

$x=x_{0}+v_{0}t+{\frac {1}{2}}at^{2}$

$v=v_{0}+at$

$v^{2}=v_{0}^{2}+2ax$

Applications of Kinematics

Motion

Motion is by far the simplest application of Kinematics, and uses the formulas with no adaptation.

Gravity and Falling Objects

Gravity affects most all earthly motion, and therefore is discussed first. Gravity pulls all objects the same distance from the earth with equal acceleration, regardless of mass, as discovered by Galileo in his famous test on the Leaning Tower of Pisa. In this section we will assume that friction and air resistance are negligible. The standard gravity constant, 'g', is the is equal to 9.8m/s², and is the acceleration done by the earth's gravity at sea level. Therefore, a falling object fall accelerating at rate g. When using this with the Kinematics formulas, g replaces a.

Example 1

Show that the force (F) of a body is directly proportional to its momentum