# Kinematics/Particles

Particle kinematics is the study of motion of particles without any reference to the causes of their motion. Though the idea behind this book seems very intuitively clear, the term motion (and rest)must be precisely defined qualitatively and quantitatively without any ambiguity. This and other common topics of Kinematics will constitutes this book.

The work assumes a good grasp of Mathematics. A small primer for the same might be written at a later date. Readers with limited mathematical background will not find this book enjoyable.

The abundance of useless jargon in modern Physics texts is the prime reason for writing this text. Modern writers of Physics seem to have got an idea that filling the lessons with quotations of great people, interesting anecdotes, unnecessarily fancy drawings, etc sell a book. Perhaps they are not wrong. If the reader is in search of such a book, then this isn't it. It is quite plainly, very distracting, to have so many side issues in a text.

In this book, historical references, anecdotes and other trivia, will be reserved for the appendix. The thrust of the development period will be on developing clean and crisp chapters which bring out cleanly the ideas that are central to Particle kinematics.

A set of problems is also intended to be developed.

## Position

Central to the study of Particle kinematics, is the idea of position. The position of a particle can be said to have three distinct specifications:

1. The point from which the position of the particle is specified, called the Reference Point.
2. The distance of the particle from the reference point.
3. Some quantifier of the direction in which the particle lies with respect to the reference point.

It can easily be seen that if any one of these three specifications aren't mentioned, the description of position is incomplete.

The position of a particle can be specified via various artifacts that have all these three specifications. The most common of these are vectors and complex numbers.

### Position Vector

The position vector of a particle is the vector drawn from the reference point to the particle. If the reference point is $O$ and the particle's position is at $P$ , then ${\vec {OP}}={\vec {r}}$ is the position vector of the particle $P$ .

## Motion and Rest

If the position vector of a particle (with respect to some arbitrary reference point) changes with time in a given time interval, then the particle is said to be in motion in that time interval with respect to the reference point. However, if the position vector is invariant within the time interval considered, the particle is said to be at rest with respect to the reference point.

Rest and motion are attributes of a particle that are solely dependent on the reference point. A particle can be in rest relative to one reference point and in motion relative to another. Thus, rest and motion are combined properties of the particle and the reference point.

Further, it can be said that if a particle is in motion with respect to a certain reference point, then the particle's position vector is a non constant function of time. Motion of a particle causes the position vector to change in both magnitude and direction in the general case.

## Path

The locus of end points of the particle's position vector is said to be the path of the particle.

## Distance

The length of the path curve traced by a particle during a time interval is the distance traveled by the particle.

For a particle in motion the distance traveled is always an increasing function of time. Hence, if $s(t)\,\!$ be the distance traveled by a particle from a fixed point on the path, then, ${\dot {s}}(t)>0\,\!$ .

The dot over $s(t)\,\!$ denotes differentiation with respect to time.

## Displacement

The displacement of a particle between two positions occupied by the particle at different time instants is the change in position in that particular time interval. If ${\vec {r}}_{1}={\vec {r}}\left(t_{1}\right)$ and ${\vec {r}}_{2}={\vec {r}}\left(t_{2}\right)$ , then the displacement in the time interval $[t_{1},t_{2}]$ is