Interpretation of Kinematics: Unraveling the Relationship Between Kinematic Variables with Graphical Analysis

 A fundamental challenge encountered by students when employing differential equations in physics problem-solving lies in the lack of foundational understanding regarding the interplay between variables and their behaviors. To elucidate the application of differential equations in physics, let's begin with kinematics, specifically examining scenarios featuring constant acceleration.

In discussions of kinematics, we traditionally delineate distance, velocity, and acceleration as distinct entities. However, they are intricately intertwined, with distance serving as the cornerstone and ultimate objective of the other two variables. Let's demystify how these seemingly disparate variables converge around the central concept of distance.

Let me explain how these seemingly different variables all relate to the same concept: distance. there's no mystery there. Distance is according to Oxford Dictionary, the space between two objects. Now, what happens if that distance changes over time? This question may look a little bit out of context, why are we even interested in moving objects?

Why would we be interested in statics objects? What mystery remains for something remains the same over time? The interesting part comes when objects move, when they change. Why? Because we can use this behavior to predict the future and the past of the object, and this can be extended to a practicality of all kinds of stuff with moving objects, like work for example. 

Explained this little insight of why moving object are important, let’s keep forward. We're then discussing the 'change of distance over time,' (or the change of a object over other kind of unit different of time, I recommend read the following article CONVERSIÓN DE UNIDADES - TIEMPO to understand what I am talking about and some deep understanding of the units of measure) which is essentially the measure of how far we'll move after a certain time interval. This concept leads us to velocity, measured in units of m/s. Furthermore, if this velocity is also changing over time, meaning the rate of change of velocity is not constant, then we're dealing with acceleration, measured in units of m/s².

Application --> Predicting future behavior and understanding past events.

There's no point in observing objects in an instant moment; when studying a phenomenon, we want to observe its evolution over time, comparing it with other factors (as mentioned in the article on units of measurement and reference points). In most scientific scenarios, we study how an object evolves over time, observing its different phases within a time interval. For instance, if I'm studying the moon's movement around the Earth and measure its distance from a certain point over time, I can deduce how much the moon moves within a specific time interval. This data is valuable because I can extrapolate it to predict the moon's future position or where it was in the past.

Thus, information about distance alone is not sufficient for predicting any kind of behavior; we must study the object's behavior in relation to time, which is what we understand as 'velocity.' And why do we do this? Simply to determine the object's distance or position at any time interval. Similarly, when we discuss acceleration, it's ultimately to measure distance or position. Therefore, although we may perceive distance, velocity, and acceleration as three distinct variables, they are essentially interconnected, with velocity and acceleration serving as units that provide us with the rate of change in distance over a given time interval, ultimately helping us determine position.

How do we determine that position? If we know the velocity, and we want to know the position at certain time interval, we multiply that velocity for the time interval of interest, and then we obtain the position value.

Now that we've introduced the topic, let's explore some scenarios to better grasp the relationship between distance, velocity, and acceleration.

First Scenario:

We have a static object. If we make a graph of distance versus time, it will look like this:

The y-axis represents distance, and the x-axis represents time. We see a horizontal line. What happens if we try to calculate the rate of change of the distance over time? In other words, what would be the slope?

Firstly, let's clarify what slope represents:

Slope indicates the rate of change; specifically, how much distance is covered in a given time interval. This concept is crucial as it provides insights into the object's motion. When we divide distance by time, we get velocity, measured in units of m/s. Hence, in a distance-versus-time graph, the slope directly corresponds to velocity.

Now, revisiting the initial scenario featuring a horizontal line: 

Upon analysis, we observe that the distance remains constant across all time intervals. Consequently, the object maintains its position without any displacement, indicating zero velocity. This insight aligns with our calculation using the slope formula.

So, returning to the first scenario, we have a horizontal line:

As demonstrated, irrespective of the chosen time interval, the velocity remains zero. Therefore, in the first scenario, we encounter a static object with zero velocity.

Second scenario:

Let’s now look at this graph of distance over time:

The y-axis represents distance, and the x-axis represents time. We can see that now the position of the object is changing over time; it is increasing. So, at first sight, we can see that if the object is changing its position, this means that there is a rate of change, or in other words, there must be a slope.

If we calculate the slope of this scenario, we obtain:

We see now that the slope is not zero, and it makes sense because for the object to move, it needs a velocity. In this case, we are moving 1 meter per second.

Now, let's delve into another analysis. Considering that the initial graph depicts distance versus time, let's explore a graph illustrating velocity over time. What would this graph reveal? Upon examination, we observe that velocity remains constant across all time intervals. Calculating the slope at any given point on the graph consistently yields a value of 1 m/s. Hence, we can deduce that the velocity remains unchanging throughout the entire time span.

This constancy in velocity signifies that the object maintains a steady pace over time. Consequently, a graph plotting velocity against time would present as a horizontal line, indicative of the consistent velocity observed. Below is an illustration of how such a graph would appear:

As we can see, the graph is a horizontal line, which means we will always have a velocity of 1 m/s over all time intervals; the velocity is constant. Therefore, when we have an object that is moving at a constant velocity, we obtain these graphs:

Both graphs are different but give you information about the same object. One graph is the study of the distance versus time, while the other graph is the velocity over time. Now you may be wondering, if we can make a graph of the slope of the distance versus time graph and obtain a new graph where we see the velocity, can we do the same with the velocity graph? Can we make another graph with the slope of the velocity graph? Sure, we can! That slope will be the acceleration, and it will be the third scenario.

Third Scenario:

Having explored the graphs of a static object and one moving at a constant velocity, let's delve into the graph of an object experiencing non-constant velocity, i.e., one whose velocity changes over time.

In this scenario, the distance-versus-time graph cannot maintain a constant horizontal line, nor can it remain a straight line, as the object's velocity varies. What does the graph of an object with changing velocity look like? It resembles the following illustration:

As evident from the graph, calculating velocity or slope at different points yields varying values due to the non-constant velocity. Consequently, the distance covered also fluctuates over time, reflecting the dynamic nature of the object's motion.

You can pick different points and find out that the slope of this graph is changing. Remember that the slope of a graph of distance versus time is the velocity; therefore, the velocity is not the same at every point. Now, let's see how this velocity versus time graph looks like:

Upon examining the graph depicting velocity over time, we notice a consistent linear increase in velocity, indicating a change in velocity over time. But what does the slope of this velocity-time graph signify? As many of you may intuit, the rate of change in velocity corresponds to acceleration. Therefore, the slope of a velocity versus time graph represents acceleration. In this scenario, the acceleration remains constant, as evidenced by consistent slope values across the graph.

Consistently, at each point on the graph, the calculated slope yields the same value of 2 m/s², confirming the constancy of acceleration over time. With this understanding, let's visualize the graph of acceleration versus time, which would appear as follows:

Which now makes sense, since we know already the acceleration is constant. Therefore, it will be the same over all time intervals, so it will be a horizontal line. And if we try to calculate the slope of this acceleration versus time graph, we will obtain zero because it is not changing.

Therefore, in an object moving with constant acceleration, we have that the distance is not the same over all time intervals. To be precise, the behavior will be a quadratic behavior, while the velocity is also changing but at a constant rate. So, it will behave as a linear graph, and the rate of change of the velocity, or slope of the velocity is not changing at all, and thus the acceleration is a constant and therefore is a horizontal line.

Summary of Graphs and Slope relations:

-Graph of Distance versus time, is (m/s), the slope is the Velocity.

-Graph of Velocity versus time, is [(m/s)/s] or (m/s²), the slope is Acceleration.

-Graph of Acceleration versus time, is [(m/s²)/s] or (m/s³), the slope is the change of acceleration over time, but we are not going to talk about it in this post, we were dealing in the third scenario with a constant acceleration.