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Newton's Laws of Motion are vector differential equations. The most common coordinate system used for their expression is Cartesian coordinates (OXYZ) with origin at the surface of spherical Earth and the positive Z-axis vertical upward. This space has three two-dimensional subspaces, OXY (with Z-symmetry), OXZ (with Y-symmetry) and OYZ (with X-symmetry). For the purposes of study the perspectives "OXZ" or "OYZ" are alike: gravity acts in the Z-coordinate. Interestingly, when one views physical reality and occurrence choosing the horizontal, OXY, two-dimensional plane, gravity is excluded.
In this section we will separate the equation into its three components. The resulting scalar differential equations are those commonly presented in HS physics texts where they are solved "in combination" for solutions of projectile motions. Before that is done, lets restate the Laws of Motion - vector form.
Newton's Second Law applies "as time proceeds" that is, during the times of motion of a BODY  in 0XYZ-space.  Newton believed all motion had a common explanation, that motion "obeyed" some physical law. Differences in motion happened in accord with different masses, momentums and forces applied. Furthermore, motions of bodies can be different because they "start differently." Newton's Second Law is written in equation form below left.
Call this Newton’s Laws of Motion. It is the Second Law,
it contains the First Law, and the Third Law
is not about a BODY nor about motion.
The equation applies at time equal to "0+" and thereafter.
At start the position was Po and the velocity was Vo.
Equation (1) is the commonly written mathematical representation of Newton's Laws of Motion (First and Second). The equation is a first order, vector differential equation with the momentum of a BODY as its dependent variable and time as the independent variable. Equation (1) contains the calculus expression "d[mV(t)]/dt." This term is called the "time-rate of change of momentum; the equation is sometimes referred to as being a "rate form."
The usual case in texts is that this Equation (1) is written by itself which is "with insufficient information" for solution. The group, Equations (2), state the time domain of the general event as being all times after commencement of observation, t > 0+. Initial conditions of the BODY are its position (Po) and velocity (Vo). To be complete and sufficient for a solution both Equations (1) and (2) are needed; they are a set.
Proceeding from Equation (1) our first step is to expand the "summation of forces." Newton categorized forces as either "acting from a distance" or "acting directly" on the BODY. Gravity acts at a distance, one notation for forces that act directly is to subscript them "app" meaning "applied" as by some agent of the surroundings.
The gravity force is often simply called gravity. We write it as Fgrav. While there might be a number of direct forces, we identify them as only one (or the resultant of all others) as Fapp where "app" means "applied." The component forms will be an "x" component, "y" component and a "z" component. The "x" and "y" components will be identical upon change of the letter "x" to "y" (or the reverse). Thus, to reduce writing, we reduce the space to 0XZ. Reduced forms of initial conditions (2) apply; these will be written upon conclusion. Our vector equation ready for reduction to two scalar equations becomes:
The above is a vector differential equation in 0XZ-space. The velocity of the BODY, the gravity force and the (sum of) applied force(s) written in component form are:
Substitute these into the equation.
The above equation spans the 0XZ-Space. It has X and Z components which are association with its I and K vector directions. We can select the 0X and 0Z parts of the equation by vector scalar multiplication of the above by I then K, respectively.
The X-component obtains by vector scalar multiplication by the unit vector, I:
The Z-component is obtained by scalar vector multiplication by the unit vector, K.
These operations are easy to understand. Notice the manipulation of form can be reversed. Simply multiply the result above by K, the result above that by I, then add them to obtain the equation at the top of the page. To be prepared for what follows, commit these steps to memory. While the vector form is preferred for investigations of motion, an exception arises soon when kinetic energy, potential energy and work of a body are explained.
 The subscript notation "BODY" means the system, the physical entity being studied is modeled as a body. The mass of the model "BODY" resides at a point.
 Matter exists and moves in our space which has three dimensions. Components of the 0XYZ space are the 0XY, 0XZ and 0YZ spaces. An event occurring in the OXZ-Space is the same as in the 0YZ-Space: just change X to Y.
 The equation applies to a BODY in 0XYZ space, in general. To apply it specifically requires the specification of two initial conditions which are the initial position and velocity.
 There are three body forces or "forces at a distance." Gravity, electrostatic (that of a stationary charge at a distance) and electro-magnetic ( that of a moving charge at a distance).