If you wish to make an apple pie from scratch, you must first invent the universe
In a similar manner, if we wish to build a Solar System from scratch, we must first start with the fundamental concepts and building blocks. Short of inventing the universe, we will start with the study of motion. Namely, we will start with the mathematical tools that we use to describe the motion of a body.
To more easily describe the motion of objects that we are studying, we simplfiy the object by imagining that the object - regardless of its size or shape - is a tiny particle that we treat as a single point. For describing the position in one dimension, we imagine that the particle is sitting in a coordinate system, where the position of the particle can be anywhere along a line that we call an axis, with respect to an arbitrary point that we call the origin. The origin is considered the zero point, and the number of units from the origin to the object on the axis is used to represent the position of that point on the axis.
In two dimensions, we simply add another axis. Typically, they are called the x and y axes. If we wanted to track the position of an object in three dimensions, we could add another axis. For simplicity, we will focus on two dimensions for now, but many of the same concepts can easily be extended to three dimensions.
In physics, the Cartesian coordinate system is used to specify the position of objects in space over time. We can directly plot the motion of an object in two dimensions on a coordinate system, where the x and y axes are perpendicular to each other and each represent a plane of motion.
Motion, however, implies that time is moving forward. How do we represent time going forward? In the above graph, it was implied that time was increasing and that the object was moving along the plotted path as time increased. We can represent this in a graph, where in one dimension, the x-axis represents time, while the y-axis represents position with respect to the origin. At any point in time along the x-axis, there is a corresponding point on the y-axis that represents the distance of that object from the origin.
The following chart describes the motion of an object in one dimension, the very first visualization shown. Between 0 - 4 time units, the object is moving away from the origin until it reaches a posiiton of 4 units to the right. Between 4 - 8 time units, the object moves towards the origin, and we see the graph slopes downwards until it reaches position 0.
Tying this all together, we can describe the position of a particle in one dimension at a given time using a single function - called the position function - that we represent with the symbol \( p(t) \), where \( t \) refers to the time, and \( p \) refers to the position of the particle relative to the origin at the given time.
$$ p(t) = position $$
Looking at the above chart, we can come up with an equation that can be used to describe the motion of the object.
Between time 0 - 4, we can use the following equation:
$$ p(t) = t $$
Between time 4 - 8, we can use the following equation:
$$ p(t) = 8 - t $$
We can validate this ourselves by substituting \( t \) into any of the above equations, using the correct function for a given time, and see that we get the correct position on the graph.
Similarly, the motion of an object in two-dimensions can be described by two position functions. For example, \( x(t) \) and \( y(t) \), for the x-axis and the y-axis, respectively. We call this the Cartesian coordinate system, where the position of a particle is defined on the Cartesian plane.
If we know what \( p(t) \) is as a function, then we know everything we need to know about the motion of the particle, as we will see in later sections.
When studying the position of a particle in two dimensions, we apply the same logic, except now we are looking at position along two axes. Therefore, the motion of an object in two-dimensions can be described by two position functions. For example, \( x(t) \) and \( y(t) \), for the x-axis and the y-axis, respectively. We call this the Cartesian coordinate system, where the position of a particle is defined on the Cartesian plane.
The position of a point-like object can then be given by the co-ordinate pair,
$$ ( x(t), y(t) ), $$
or as a vector,
$$ \vec{r}(t) = x(t) \hat{i} + y(t) \hat{j}, $$
where \( \hat{i} \) and \( \hat{j} \) are unit vectors pointing in the x and y-axes, respectively. The two directions are independent of each other, as they are perpendicular axes (i.e. moving along the x-axis does not affect movement along the y-axis).
When we graph a particle on the Cartesian plane, we see the path that the particle takes over time in two dimensions. Notice that time is not represented on the graph, and it is implied that time increases as you move from the starting position to the final position.