The physics of nerf guns

A class activity is described that involves taking a video of a nerf gun fired vertically. The ascension time of the nerf bullet is found by counting the number of video frames, which is then used to calculate the maximum height reached. The kinematic equations can then be used to calculate several extra pieces of information about the trajectory of the nerf bullet such as the initial velocity and the spring constant of the nerf gun, etc.


Introduction
The nerf gun is a very useful contribution to the physics teacher's armamentarium.For example, the nerf gun can be used in teaching quantum mechanics (Kaur et al 2017), and motion experiments in the classroom (Marry 2015).Cassidy (2023) describes the use of nerf guns to increase school student engagement in STEM.
The current paper describes an idea for using a nerf gun to practically illustrate the equations of motion, energy, Hooke's law, impulse, and power.These subjects are usually taught together over a period of a week or two in an introductory physics course in high school or first year university.
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Method
The nerf gun seen being fired in figure 1 was obtained from an Australian store called Big W and cost 7 AUD, which included eight bullets, (now down to seven since one is missing high up on the ledge of a university building).

S Hughes
Figure 1.A nerf gun being fired close to a building.The photo was extracted from a SLO-MO video frame.The arrow shows the position of the nerf bullet close to maximum height.

The physics of nerf guns
About half an hour was spent outside collecting standard (20 frames per second (FPS)) and slow-motion videos (240 FPS on an iPhone 12).Timing is more accurate with slow-motion videos.When a nerf gun is fired vertically, there will always be some laterally motion, and so the bullet follows a parabolic path as seen in figure 2, although not quite as extreme.
Back in the classroom, a freely available program, Tracker (https://physlets.org/tracker/trackerJS/) was used to count the number of video between firing to the maximum height (t 1 ).The fall time (t 2 ) was also extracted by counting video frames.This time was used to calculate the maximum height and then all the other values as described below, and shown in figure 2 and table 1. (Note that not all equations in the text appear in figure 2 and table 1).Table 1 was projected onto a white board by data projector with an empty fourth column, and the values gradually entered as calculation proceeded, The PE imparted to the bullet is, GPE = mgh. (2) Eight nerf bullets weighed 8.64 g giving an average of 1.08 g.So, we can calculate the PE.
The initial velocity of the bullet is the same as if the bullet was dropped from the maximum height down to the firing level of the gun, and so equation (1) can be used to find the initial velocity, The velocity is also given by v = √ 2gh.Therefore, we can calculate the and cross check with the PE, which of course should be the same, since both values depend on the time to reach maximum height.
The energy stored in a spring is, where EPE is elastic PE, k the spring constant and x the spring extension, which is the distance the priming mechanism is pulled, 33 mm, for the nerf gun used in this experiment.
When we have got the spring constant (k), we can use this with the spring extension to calculate the maximum force accelerating the nerf bullet when the spring is at maximum extension, which should be twice the force calculated using equation ( 10), assuming that the spring follows Hooke's law.The values for equations ( 7) and ( 10) in table 1 show that this is so (5.97 and 2.99 N respectively) the height from the ascension time, we can use that to estimate the fall acceleration from the fall time (t 2 ), This is expected to be a bit less than 9.81 ms −2 due to air friction.The bullets tend to travel up vertically but come down more sideways.
Additional information that can be obtained is the acceleration of the bullet in the barrel of the gun by rearranging equation (3), In this case, s is the extension of the spring (x), and the initial velocity (u) is zero, so we can write, We can go further and use Newton's second law to find the force (F) applied by the spring to the bullet, and then the impulse (I), Figure 2. Schematic diagram of a nerf gun being fired and the sequence of equations.Equation numbers match those in the text and table 1.
The power of the spring can also be calculated from, The sequence of calculations is shown schematically in figure 2.
An extra calculation can be done to determine the force of air friction using the time for the bullet to fall to the ground (t 2 ).The fall acceleration (a 2 ) of the bullet is given by The difference (∆a) between the fall acceleration (a 2 ) and the acceleration due to gravity (g) enables the retarding force of air friction, or the drag force (F D ), to be calculated.
The friction force, or drag force, can be used to roughy estimate the drag coefficient (C D ) from the drag equation, (www.engineeringtoolbox. com/drag-coefficient-d_627.html) where ρ is the density of the fluid an object is travelling through, in this case air, ρ = 1.293 kg m −3 , v the velocity of the object relative to the fluid and A, the reference area, generally the same as the cross-sectional area of a solid object.For the falling nerf bullet, the mean fall velocity can be used in equation ( 17) to estimate the drag coefficient, vf = h/t 2 (18)

Results
The results are collected together in table 1.

Discussion
The maximum height of the nerf bullet was estimated by placing a metre rule on the ground beside the wall and taking a photo from the other side of the courtyard.The image was loaded into ImageJ and the calibration feature of ImageJ used to estimate the maximum height reached by the bullet.This gave a value of 11.2 m which is 11.2 − 9.3 = 1.9 m greater than the height calculated from the number of video frames.The reason for this discrepancy is parallax.For example, if the bullet was 1 m from the wall and being viewed at an angle of 45 degrees, the bullet would appear to be 1 m higher.If the bullet was 2 metres from the wall and the viewing angle 45 degrees the height would be an extra 2 metres, which is the discrepancy found for the value presented in this paper.
Another interesting exercise is to calculate how high the nerf bullet would go on other celestial bodies such as the Moon, Mars, and Ceres (56.1, 24.5 and 321.2 m respectively).

Conclusion
This single demonstration brings together the three basic equations of motion, gravitational PE, S Hughes KE, elastic PE, Newtons second law, Hooke's law, impulse, power, and air friction, and is an example of a cascade of data that can extracted from an initial single value.There is a lot of scope for extending this exercise, for example error analysis, measuring horizontal range, investigating the drag coefficient of nerf projectiles, etc.The skies the limit (pun intended).

Table 1 .
Data extracted from measuring the time for a nerf bullet to reach maximum height, along with the mass of nerf bullet and spring extension of the nerf gun.The equation numbers match those in the text and figure 2.