Newton's Second Law

The study of mechanics stems almost as a philosophy of Newton's study of the natural world itself. Therefore, we can think of Newton's 3 Laws of motion as a philosohpy in and of itself—Newton had pioneered the study of the natural world, and I want to make it a challenge to myself to construct the broader equations of physics as a result of this equation.

The famous equation describes the motion of an object in relation to forces, specifically it pinpoints the role of force and what it does on an object. He first assumes a world at either two states: constant velocity or absolute rest. Therefore, he asks

F = m*a

Acceleration, Velocity, and Position

Coincidentally, acceleration, velocity, and position are intrinsically linked with the derivative operation. Additionally, you can obtain these quantities through the reverse operation of integration. These typically reverse the operations of differentiation through this integration, as in, you can obtain these seemingly three different quantities with the same operations. Below are examples of the usages of the definition. This process repeats throughout each of them, as follows.

F=m*a;a=d^2(x)/(d(t)^2)=d(v)/d(t)

v= (lim_Δ(t)→0)(Δ(x)/Δ(t))=d(x)/d(t) ;v=d(b*t^2)/d(t)=2*b*t

We will extrapolate the definitions of derivative and integral operations through the use of defining acceleration, velocity, and position. Given that integration is the opposite of derivation, we can assume that the process to move down the chain (from acceleration, velocity, and position) can also be done through integration up the chain (position, velocity, and acceleration). Therefore, the integration of position is an upward moving chain, so integration of position gives you velocity, and the integration of velocity gives you acceleration.

I have made the deliberate effort to, instead of moving up the chain, move down the chain from acceleration to position. This is because I want to work off Newton's Second Law, and therefore I assume that acceleration is the accumulative integration of position, and obtaining the velocity and position of an object requires integration. In other ws, we use the definition of the bottom to derive upward.

Vector Quantities and Scalar Quantities

We can't talk about acceleration and the usage of acceleration, or the action of force and the usage of force, without acknowledging the fact that acceleration and force are vector quantities. There is a distinction amongst vector quantities that doesn't exist within scalar quantities, in that the vector not only contains a magnitude length but a set of coordinate vectors as well. These coordinate vectors can change, but that is only if the coordinate system changes.

A=(A_x)*i+(A_y)*j

(A_x)=A(cos(θ)) ; (A_y)=A(sin(θ))

Lastly, if we assume a Cartesian vector, that is, one that has components that are defined via the Cartesian coordinate system, then we can derive the coordinates of the vector quantity by using sine and cosine. Conversely, by vector addition, we can say that within the Cartesian coordinate system, the vector components simply become the vector magnitude.

If we know that force is a vector, we can say that the force can be split into component arts. This actually helps a lot because . And remembers his is because this is separated into component parts, so being able t understand the coordinate system that you're working with because defining the coordinate system means that you're able to work with the individual parts themselves. This comes in handy when you're having to do arithmetic across similar parts. Remember that we want to do