Investigating quadratic functions by analysing the projectile path of a paper rocket
I am going to explore quadratic functions and create a parabola for the path of a paper rocket. The rocket used was 30 cm in length, with hot glue in the nose-cone and 4 equidistant fins. The variables used will be h for the height and t for the time, in the function.
Seen below is the image after the rocket landed, the blue dots representing the front tip of the rocket in each frame. The placement of the dots was done using Logger Pro.
In this image, we can see that the rocket made a projectile motion and landed approximately 33 metres from the launch area. The corresponding graph (graph-1) is shown below.
In the graph, we can see that the rocket follows a projectile motion and it points to make an upward arc, meaning that the function would have a negative leading coefficient. The domain of this graph is 𝑡 ε 𝑅 because the graph can be drawn in one go, and the range is h > 0 because the projectile of the rocket isn’t going underground. We can also observe the data table (table — 1) of the coordinates of each frame the rocket was in. The points are only visible till 1.5 seconds, therefore, below is a more concise table (table 2), with 2 coordinates for every second.
We can observe in this table that at 1 second the rocket was reaching the vertex, at 2 seconds it was approx. at its vertex and by 3 seconds it was descending. From this, we can conclude that the vertex of the function will be between 1 second and 2 seconds.
To calculate the actual vertex of the graph, I would have to find the best fit function for this parabola. To do that I’ll choose 3 points from the above image. For this calculation, I will choose (0.19, 3.64), (1.99, 18.59), and (3.79, 2.74). First, I will first create the equations for these 3 coordinates.
3.64 = (0.19)2𝑎 + (0.19)𝑏 + 𝑐
18.59 = (1.99)2𝑎 + (1.99)𝑏 + 𝑐
3.79 = (2.74)2𝑎 + (2.74)𝑏 + 𝑐
Now to find the equation of the line, I used polysmt2 in the T-64 Plus graphics calculator (GDC) by substituting the values for x and y, which in this case are t and h respectively. The result gave me the equation: -
h = − 10.99𝑡2 + 32.27𝑡 − 2.09
Then I proceeded to graph the equation on the GDC and find the vertex of it. To do that I had to adjust my window because the graph was too small to be seen in my previous settings. Graph — 2 shows the parabola made by the GDC.
The vertex of this function is at (1.47, 21.58). Meaning that the highest point the paper rocket was at was 21.6 metres and it was at that maximum at 1.47 seconds. When I checked my data table to compare my findings, I noticed that the actual highest point that the rocket was at was (1.99,18.59). This made me realise that my function was incorrect.
To evaluate my function further, I graphed it into Desmos with the coordinates that I got using Logger Pro, and the difference was a lot. As you can see, the black parabola is the value of the function that I calculated. The green points are the coordinates from table — 1. The function doesn’t match up with the coordinates. This made me conclude that my function was incorrect, therefore I had to rely on the algorithm to make the best fit function for me.
Therefore, t he best-fit function that it determined was :-
h = − 4.56𝑡2 + 18.07t + 1.02
When I graphed it in Desmos, the function was almost accurate but the maximum of the function was at (1.98,18.92) instead of (1.99,18.59), indicating that the algorithm wasn’t 100% accurate.
The best-fit function was more accurate than my function. My calculations themselves had no error, therefore the only other reason for that might be because my calculations were limited to only 3 points. The algorithm must have taken an average of all the coordinates present and then determined an appropriate function that would give similar results to the path of the projectile.
Other variables that could have affected the path of the projectile were speed of the wind/weather, the build of the rocket itself and the pressure that was applied to the rocket. The weather conditions when we launched the rocket were clear, with no/minimum wind blowing. If there was wind, the rocket would have angled, which would have made it harder for us to record the data. We used the same rocket throughout the exploration, meaning that the size of the rocket or the weight of it, won’t affect the results. Some limitations of calculating the path of the rocket were placing the blue dots and the calculation part itself. The quality of the recorded video was sometimes not as good, and the rocket was partially visible in some frames, moreover all the dots had to be manually placed. So, there were a few errors with the dots, that served as one limitation. As stated before, manually calculating the function wasn’t accurate at all, whereas the Logger Pro algorithm gave a better, averaged function. The only con to using that function is that the vertex is a bit higher than the actual maximum of the projectile path.
In this paper, I explored projectile motion and creating a function for the launch of a paper rocket. There were some limitations because of which my function wasn’t accurate, hence I chose to use the Logger Pro function h = − 4. 56𝑡2 + 18.07t + 1.02 that determined the path of the function. If I had more time, I would have looked more into different rocket types and to what extent did they have an effect on the path of the rocket. I would have changed the sizes, the weight, the length of the nose cone, the number of fins, etc to help me better understand this scenario. Overall, this paper made me learn and understand more about quadratic functions and using digital tools like Logger Pro to improve my knowledge.
