Determination of the equation of a parabolic shot and calculation of the best angle for its execution

in StemSocial7 months ago (edited)


Greetings again my dear friends I hope you are all very well, we continue linking us with the wonderful phenomenon of movement in its different trajectories as is the case of parabolic motion and, of this movement we will analyze this time the generation of the equation of a parabolic trajectory described by the launch of a basketball, but also calculate the possible angles of shot to be able to put the ball in the basket.

How many of these types of parabolic movements can we see around us, surely many, that's why this time we move to basketball where you can see the beautiful parabola described by the ball being thrown by the players to the basket, therefore, physical science and mathematics can be used everywhere where we are, then my dear friends to continue expanding our knowledge in relation to the phenomenon of movement, especially the parabolic, then I leave you the following approach or statement.


A basketball player makes a shot with a ball towards a basket, the player is at a distance of 6.25 meters (m) from the basket, the height at which the ball went out was 2.35 meters (m) and whose initial speed was 10 m / s, the height of the hoop that holds the mesh is 3.05 meters (m), therefore, in relation to the above answer the following questions:

a.- What is the equation of the trajectory of the parabolic launch?

b.- What is the best angle for the player to get the ball into the basket?



Vo = 10 m/s (Initial bolon velocity).
g = 9,8 m/s2. (Gravity)
X = 6,25 m (Distance of the ball throw).
HL = 2,35 m (Throwing height of the ball).
Ha = 3,05 m (Hoop height).
Y = Ha - HL = 3,05 – 2.35 = 0,70 m.
θ = ? (Launch angle).

a.- To begin answering our first question, we will start by visualizing the following figure 1.


We then identify the formulations to be used, thus, we have the following formula 1.


For formulation 1 above, we need to describe the variables that identify time, and for this we have:


Now we substitute the algebraic value of time (t) into equation 1.


In this way we obtain the formula 3, which represents the equation of the trajectory of the parabolic throw made by the basketball player observed in the gif of the presentation of this article.

b.- After constructing the equation of the parabolic trajectory of the ball, we proceed to answer our second question, i.e., what will be the best angle for the parabolic launch, and which will give the player the best chance of scoring the ball, then, we proceed as follows.

Aplicación de la ecuación de la parábola.jpg

When applying our parabolic trajectory equation we notice that we obtain a second degree equation, therefore, we proceed as follows in order to find the best angle of shot for the player.

Cálculo de los ángulos de lanzamiento .jpg

From this handle we obtain two draft angles, one of 70.22° and the other of 26.10°, so that, in this case, the best angle will be the one of 70.22° because it provides a higher launch than the angle of 26.10° can provide.


The human being performs a large number of activities, where any kind of movement is developed, especially in sports disciplines such as basketball, and we were able to prove this in this opportunity, and we were also able to link it with physical science and mathematics by determining the equation that describes the parabolic trajectory of the shot when it is thrown towards the basket.

We also managed to calculate two shooting angles by using the equation of the parabolic trajectory described above, and we analyzed how the most advisable angle is the one of 70.22° because it will allow a greater elevation of the ball towards the hoop, and with the other angle the shot would go straighter without the necessary inclination to reach the height of the hoop, a wonderful example of the implementation of parabolic motion using the physical-mathematical interpretation to find important applications in basketball and certainly in any other sport.

Until another opportunity my dear friends.

Note: The images were created by the author @rbalzan79 using Power Point and Paint, the animated image was created using the PhotoScape application.

Recommended bibliographic references

[1] Parabolic Projectiles. Link.
[2] El movimiento rectilíneo. Link.
[3] Projectile Motion. Link.


And sportspeople think math won't help them! :P

Hello, yes, mathematics is the most universal language that we know as humanity and that has given us so much interpretation on any phenomenon that develops around us, thank you for your valuable visit and comment. Greetings.

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