# Basic kinematics

Kinematics is the study of the motion of objects. Basic kinematic problems are approached using Isaac Newton's laws of motion.

## Newton's Laws

Newton attempted to explain how objects move using as few assumptions as possible. These assumptions are what we call Newton's Laws today. They are remarkable in that they have stood the test of time for almost all motion except those at the smallest scales (quantum mechanics) and the largest scales (general relativity). Even then, we have mostly just added or modified some assumptions that we had thought reasonable to assume about the nature of the universe, but the three laws have mostly remained. The laws are used to deduce the fundamental equation we will be using to study almost all of classical physics below. The wording of the laws has been altered slightly in order to be less jarring to a modern student. Another translation of the original laws can be found here.

### Newton's First Law: The Law of Inertia

Every body remains in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed thereon.
At this point, Newton has not told us what a force is, so this law isn't a description of how bodies move. You may think "Hey, I know what a force is, I force something to move when I push on it." However, that "contact force" is not the only type of force Newton means to deal with. He has basically used this statement to define what a force is: anything that changes the linear motion (including zero motion) of a body. This allows us to include gravity as a force. As a matter of fact, we will see that classical physics successfully describes all forces in terms of fields (similar to the way gravity works). Your pushing on an object is actually a result of electromagnetic field interactions between the particles in your hand and the particles in whatever you're pushing. Your hand never actually comes into direct contact with anything, in the sense of there being no space between your hand and another object.

### Newton's Second Law: The Law of Acceleration

The change in quantity of motion is proportional to the magnitude of the force; and is made in the direction of the straight line in which that force is directed.
This law is extremely rich in content, as Newton used devices similar to what we refer to today as Newton's cradle to note that the "quantity of motion" given to the end ball (assuming perfect collisions where all the "motive force" of the ball you lift on one end is transferred to the ball at the other end) is proportional to the final velocity of the ball you initiate. He also noted that different types of balls imparted larger arcs to other balls and thus he came up with the product of mass (a quantity proportional to the object's weight) and velocity as being the "quantity of motion" of an object that should be preserved in perfect collisions. Today, we call this product the momentum of an object and usually denote it as
$\mathbf {p} =m\mathbf {v}$
where p and v have both a magnitude and direction, as described by Newton's laws. Manipulating objects that have both magnitude and direction (called vectors) is covered by vector algebra, which will be talked about in a supplementary review guide. Now, if we denote the force vector by ${\vec {F}}$, then Newton's second law tells us that
${\vec {F}}={d{\vec {p}} \over dt}={\frac {d}{dt}}(m\cdot {\vec {v}})=m{d{\vec {v}} \over dt}$
The ${\frac {d}{dt}}$ is just the ordinary time derivative from calculus. If you have not yet studied calculus, the above can also be written as
${\vec {F}}=m\cdot {\vec {a}}$
where a is the change in velocity with respect to time, also known as acceleration.
* Exercise 1: If you know a bit of calculus, prove that if the total force on a system of two particles (or pool balls if you prefer) is zero, then momentum is a constant. Note that this is the case when two pool balls hit each other (assuming idealized circumstances). We say that momentum is conserved.

### Newton's Third Law: The Law of Interaction

To every action there is always opposed an equal reaction; or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.
This law is primarily for idealized objects such as spheres and points that act on each other's centers of motion. However, it is the impetus that allowed us to separate mass from weight. In symbolic terms, it states that if one body exerts a force ${\vec {F}}_{1}$ on another body, then the second body exerts a force ${\vec {F}}_{2}$ on the first body, and in vector terms ${\vec {F}}_{1}=-{\vec {F}}_{2}$. Using the equation we derived from the second law, it is not hard to see that the mass can then be measured by defining one mass to be a standard, and using the ratio of accelerations as the mass of the second object.

## Vectors

Vectors are Physical Quantities that have both magnitude and direction. Some example of vectors are 'force', 'velocity', 'acceleration', etc.

## Free Body Diagrams

Free body diagrams are extremely useful tools when analyzing a physics problem. When approaching a kinematics problem, the first thing you should do is draw a free body diagram. What is a free body diagram? It is a simple sketch that shows all of the forces acting on an object ( a "free body"). When drawing a free body diagram, first you must isolate the object in question. Then, you must identify all of the forces acting on it. Remember, if an object is at rest, the vector sum of all forces must be zero. Newton's laws also tell us that every action has an equal and opposite reaction. If an object is sitting on a table, its weight is pushing down on the table, but the table is also pushing up on it (this force against the weight of an object is called the normal force. It is a very important concept in kinematics). Since both forces are equal and opposite, the object neither falls through the table nor flies up into the air.

### The Motion of a Particle

We are now going to apply Newton's laws to an idealized object, a single particle. A particle in classical physics has a very well-defined meaning: it is an object without any spatial dimensions; mathematically it is a single point. We study these simple objects first, and later we consider larger objects as being made up of a lot of particles. We will see that spheres (objects like pool balls) behave a lot like particles (classically) and so do objects that interact along their centers of mass, and thus we will talk about sphere-like objects, simple center-of-mass interactions, and particles freely. We will show that the sentence above is accurate later when we have developed the tools and familiarity with physics to do so.
Now, take a ball and hold it in your hand some distance above the floor. If you stop holding the ball, the ball will fall to the floor. According to Newton's law of inertia, the ball could not have done so unless it was acted upon by a force (inertially, it should have remained where you were holding it). Since you did not force the ball downwards, some other force besides yourself must have acted on the ball. Due to our experience with other objects falling to the ground, we hypothesize that the Earth is exerting a force on your ball. According to Newton's third law, the ball must also exert a force on the Earth of equal magnitude and opposite direction.
Derivation of basic kinematics being written....