General Mechanics/Fundamental Principles of Kinematics

The fundamental idea of kinematics is the discussion of the movement of objects, without actually taking into account what caused the movement to occur. By using simple calculus, we can find all of the equations for kinematics. To simplify the learning process, we will only consider objects that move with constant acceleration. For the first few parts, we will also assume that there is no friction or air resistance acting on the objects.

Straight Line Motion (SLM)

The name of this section Straight Line Motion means that we begin learning about the subject of kinematics by observing motion in one dimension. This means that we will only take one axis of a 3D $(x,y,z)$ coordinate system into account. We will be using the $x$ axis as our axis of motion.

Throughout our discussion, we will look at the motion of a rigid body, one that does not deform as it moves. We idealize the rigid body by assuming that it has no dimensions and is infinitely small. This way we can talk about the entire body instead of saying, "The front of the body is at this point while the rear of the body is at some other point". Also, subscripts on the variables in our equations will indicate the initial value, $i$, and the final value, $f$, of that variable.

Displacement

To start off with we will define the term displacement. Displacement is basically the shortest route of getting from one place to another, basically it is a straight line from the starting point of motion to the ending point of motion. No matter how much movement takes place in between and what sort of things come inbetween, we only care about the first location and the second location. We will use the variable, $x$, to stand for the locations of the rigid body that we are discussing. It is a vector quantity i.e. it has both magnitude and direction, if we want to define displacement for some movement then it would be like - '50 km due North'.

In order to define motion we first must be able to say how far an object has moved. This is done by subtracting the final value of the displacement by the initial value of the displacement. Or, in other words, we subtract the initial position of the object on our coordinate system from the final position of the object on our coordinate system. It is necessary to understand that the values which you receive for this variable can be either positive or negative depending on how the object is moving and where your coordinates start:

$x=x_{f}-x_{i}\,\!$

Velocity

The term velocity, $v$, is often mistaken as being equivalent to the term speed. The basic difference between speed and velocity is that, velocity is a vector quantity, whereas speed is a scalar quantity. The term velocity refers to the displacement that an object traveled divided by the amount of time it took to move to its new coordinate. From the above discussion you can see that if the object moves backwards with respect to our coordinate system, then we will get a negative displacement. Since we cannot have a negative value for time, we will then get a negative value for velocity (Here the negative sign shows that the body has moved in the backward direction to the coordinates system adopted). The term speed, on the other hand refers to the magnitude of the velocity, so that it can only be a positive value. We will not be considering speed in this discussion, only velocity. By using our definition of velocity and the definition of displacement from above, we can express velocity mathematically like this:

${\bar {v}}={(x_{f}-x_{i}) \over (t_{f}-t_{i})}$

The line over the velocity means that you are finding the average velocity, not the velocity at a specific point. This equation can be rearranged in a variety of ways in order to solve problems in physics dealing with SLM. Also, by having acceleration as a constant value, we can find the average velocity of an object if we are given the initial and final values of an object's velocity:

${\bar {v}}={(v_{f}+v_{i}) \over 2}$

These equations can be combined with the other equations to give useful relationships in order to solve straight line motion problems. For example, to find the initial position of an object, if we are given its final position and the times that the object began moving and finished moving, we can rearrange the equation like this:

$x_{i}=x_{f}-{\bar {v}}(t_{f}-t_{i})\,\!$

Acceleration

The term acceleration means the change in velocity of an object per unit time. It is a vector quantity. We use the variable, $a$, to stand for acceleration. Basically, the sign of acceleration tells us whether the velocity is increasing or decreasing, and its magnitude tells us how much the velocity is changing. In order for an object that is initially at rest to move, it needs to accelerate to a certain speed. During this acceleration, the object moves at a certain velocity at a specific time, and travels a certain distance within that time. Thus, we can describe acceleration mathematically like this:

$a={(v_{f}-v_{i}) \over (t_{f}-t_{i})}$

Again, we can rearrange our formula, this time using our definition for displacement and velocity, to get a very useful relationship:

$x_{f}=x_{i}+v_{i}(t_{f}-t_{i})+{1 \over 2}a(t_{f}-t_{i})^{2}$

Motion with Constant Acceleration

This is the simplest kind of accelerated motion. The velocity changes at the same rate for entire duration of the motion. The displacement-time graph for motion with constant acceleration is always a parabola.

The equations of motion with constant acceleration are:

$v_{x}=v_{0x}+a_{x}t$

$v_{av}=v_{0x}t+{\frac {1}{2}}a_{x}t^{2}$

$v_{x}^{2}=v_{0x}^{2}+2a_{x}(x-x_{0})$

$x-x_{0}={\frac {1}{2}}(v_{0x}+v_{x})t$