Calculation of the distance traveled by an arrow on impact with a target

in StemSocial2 months ago


Greetings again my dear friends I hope you are all very well, we continue analyzing various examples where the phenomenon of sound is involved, and thus continue to verify that this phenomenon is an intrinsic part of our existence and everything around us, which is why the sound has great utility in our activities, including sports.

Sport in its different disciplines is of great value and usefulness for the human being and, where science plays and will play a great role, much has been advanced at the level of technology applied to sport, so we can say that technology is applied science, therefore, in order to further deepen the importance of sound in our lives, we will analyze another example, through an exercise as you can see below.


In an archery competition, which consists of getting as close as possible to the center of a target when launching an arrow as you could see at the beginning of this article, for this competition the value of sound propagation in the air is 340 m/s, once the shot is executed, the person who made the shot hears a sound emitted by the target at the moment of the impact of the arrow with it, this sound takes 0.2 seconds to be perceived by the person who made the shot, and also, the judges report that the arrow takes 4.3 seconds to hit the target, therefore, we want to know the following characteristics:

a.- What is the distance of the target from the person doing the archery?

b.- What is the velocity of the arrow as it travels?



d = ? (Distance of the target from the person taking the shot).
Vm = 340 m/s (Velocity of the propagation medium of the emitted sound).
T = 0,2 s (Travel time of the sound waves emitted by the target at the moment of the impact of the arrow).
t = 4,3 s (Time implemented by the arrow in its path)
Vf = ? (Arrow speed).

a.- First, we analyze the dice we have and, based on them, we locate the formula that suits our requirement, which in this case is to know the distance of location of the target with respect to the 'person who performs the archery, or we could also say, the distance traveled by the arrow, therefore, we have the following formula.


From this formula we can clear our required unknown, i.e., the distance, so we will obtain the following formula.

Fórmula 2_Y_Cálculo de la distancia.jpg

In this way we can determine the distance that exists between the target and the person executing the arrow shot, by means of the sound emitted by the target at the moment of impacting the arrow with it, therefore, this is the distance that the arrow had to travel.

b.- Now, once the distance traveled by the arrow is known, we will calculate its velocity, and to do so, we will implement the following formula.

Fórmula 3 y cálculo.jpg

By substituting the values we have in the previous formula we obtain the speed of the arrow with which it travels the 68 meters that exist between the person executing the shot and the target which is the receiver of the shot.


Sound has allowed us to understand nature and, strategically, we have been able to transfer this knowledge to any of our activities, as we did with the calculation of the distance between the person who made the arrow shot with the bow and the target, and this distance helped us to know the speed of the arrow, of course, with the implementation of technological tools that manage to capture the different characteristics of sound.

Undoubtedly, there are many applications of sound when it propagates from one side to another, and that both our ears or any electronic device, can capture for a particular purpose in favor of the development especially of any of our various activities, and that also managed to solve through the application of mathematical formulations.

Until another opportunity my dear friends.

Note: The images were created by the author using Power Point and Paint, the animated gif was created using PhotoScape.

Recommended Bibliographic References



[3] Echo-based measurement of the speed of sound. Link.

[4] Propiedades de las ondas sonoras . Link.


I wanna share this with my child who is now in junior high school. Thank you for your informative blog. I learned a lot.

Hello, thank you for your words and for your important visit to my blog, I hope it will be useful for your son. Greetings.

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Very interesting example to calculate it


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