Let us first recall the simplified definition of Physics: “The study of the motion of subjects,” where “motion” means the change of position after a period of time. However, it seems that in real life, when talking about “motion,” not only are we concerned about how the object changes its position with respect to time, but “how” it changes is also a concern. For example, to ride a motorcycle safely in traffic, we need to know when a car will switch lanes (or, at least, predict), how it will switch lanes (is it a 4-wheel steering car that moves relatively sideways, or is it two front-wheel steering that will have its head moving first)? Now, how can we predict “when” a car will be in our lane, supposedly we know the car will be in the lane in the future at an unknown moment, while we are still at the present.
Obviously, at this point, we are dealing with a very abstract idea of time, because what is “before”? What is “later”? What is “future”? To avoid diving into Philosophy (again, it is really easy to start questioning these concepts and accidentally do Metaphysics instead of Physics itself), “time” has to be somewhat “simpler” and less abstract so that “future” and “past” (all “parts” of “time”) can be easily defined. Let us recall the property of time: it is a parameter to which we compare things to themselves accordingly (which means “time” has to have different “selfs” so that each state of object can be attached to an individual “self” without two states exists at the same time. Note that the “two states cannot exist at the same time” here is another axiom in our model), and it has an “order” (i.e. future and past). There is a very simple Mathematical tool we can use here that satisfies both of those properties (i.e. different “selfs” that are in order): the number itself. For example, 1 is different from 2 (difference), and 1 is “before” 2 (order). Mathematically speaking, we are saying that time is a scalar in Classical Mechanics. Now, we are able to simply define what is “future” and “past” in Physics: “future” is the “version” of an object that its state is attached to a greater value of time, and vice versa for past. When a thing is not subject to change as time “moves” through its different “value”, it is said to be homogeneous in time.
Back to our questions at the beginning, how can we predict when a car will be in our lane in the future? Without any other information besides the current position of the object at the current time, it is definitely really hard to predict when the car will be in our lane. Let us assume we know how it will change according to time, as in the “process” that it will use to change its position (not the final position). Then, we can, Mathematically speaking, calculate the position of the car at each given time, and we can solve for when the car will be in our lane. This “how the car changes its position with respect to time” is called velocity. Notice that, however, “how” in this case is very narrowly defined. At this point, we are still being very abstract about what is “velocity.” Let us make it easier to visualize by using examples, again: when a car is it its lane, “how” it will change its lane refers to “how” the car move to change its lane. That is, the car has to move to the right, given its lane is on our left, to merge into our lane. There are a few “properties” we can notice: it takes time to merge into our lane, the car might signal and slowly move to the lane line, and then quickly move into our lane when it is safe to do so. Vaguely, that is the “how” we are talking about. A simpler definition would be: velocity refers to how “fast” things change their positions. Again, “fast” is not yet clearly defined, so let us give it a simple definition: if a process takes a shorter amount of time, that is the time that the end of the process is attached to is not “far” away from the time that the beginning of the process is attached to (compared to another process), it is said to be fast. Mathematically speaking, it means that the subtraction of the value of time at the end of the process and the time at the beginning of the process is a small number (compare to the subtraction of time of another process), it is said to be fast.
However, what if how the car changes its position can also change with respect to time? That is, what if velocity can also change with respect to time (as we can see in the example where the car moves slowly to the lane line)? If this happens in real life, we can very easily imagine the situation gets “chaotic”: the car will merge into the lane at an unpredictable time compared to the time in our prediction. For example, if the car was merging slowly at first, we would expect it to take, say, 10 seconds to completely merge into our lane (and if we were a reckless rider, we would speed up to pass the car before it merge into the lane). However, suddenly, in the middle of the merging process, the car speed up (which means its velocity increases) and complete the process in 7 seconds? If we were passing, we would get hit (supposedly we do not react in time). Therefore, to “completely” predict the car’s motion, we would need to know, in addition to what we know about it so far, how its velocity will change, and let us call this acceleration.
Again, what if the acceleration can also change with respect to time? Then, again, we would need to know how it changes with respect to time. If we repeat this question again and again, we would find ourselves stuck in the loop of “knowing” how the car changes its state again and again. Thus, to completely and surely predict a car position at a certain time, we would need to know all its “how-s”. Mathematically speaking, this means that the position function of the car is, in Classical Mechanics, a smooth function.
So, what exactly is velocity, Mathematically speaking? Let us define one more thing about velocity: the higher the velocity, the faster a motion happens (in this case, “motion” is the “process”) and vice versa. From the definition of “fast”, we can see that if an object’s velocity is increased when the distance is increased and the time is kept the same, meanwhile the velocity will decrease as time increases and the distance is kept the same. Mathematically, it is equivalent to saying the following:
v ∝ d
and
v ∝ 1/t.
Perhaps the simplest Mathematical relation that has this property is the following:
v = d/t
Since our model uses Cartesian coordinates to represent the general idea of “space,” the idea of vectors is necessary. The position of an object can be represented using a vector, and since time is a scalar, velocity is a vector.
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