Kinetic energy density

Kinetic energy density, more commonly KED, is a measure of how safe a dart is. KED is defined as follows:

<math>E_k^{} = \frac{E_k}{A} = \frac{\tfrac{1}{2} m V_m^2}{\tfrac{\pi}{4} d^2} = \frac{2 m}{\pi}\left(\frac{V_m}{d}\right)^2 = \frac{4 E_k}{\pi d^2}\,\!</math>

<math>E_k^{}\,\!</math> is the kinetic energy density.

<math>E_k\,\!</math> is the kinetic energy.

<math>A\,\!</math> is the minimum impact area.

<math>m\,\!</math> is the dart mass.

<math>V_m\,\!</math> is the dart muzzle velocity.

<math>d\,\!</math> is the diameter of the dart.

KED justification

Intuitively, the safety of a dart depends on the dart's mass, the dart's muzzle velocity (the maximum speed the dart will move at unless it is shot down a tall cliff or something similar), the minimum impact area, the hardness of the dart tip, and the thickness of the dart tip. Heavier darts are less safe; they are known to hurt more. Faster darts are known to hurt more. Intuitively, darts with pointy tips are known to hurt more and can readily penetrate flesh. Harder tips hurt more than soft tips.

The KED formula fits out intuition except with regard to tip softness. KED says nothing about the softness of the tip. A different dynamic analysis must be used in that case. This analysis will be detailed later. Note that a class in mechanical vibrations will provide more than enough information to characterize soft tips.

Typical KED values

The table below is based on data from Klaviel posted by Daniel Beaver at NerfHaven. Slug darts were used; a dart mass of 0.9 g was assumed in the KED calculations (#6 washers were used in the slugs).

	<math>V_m \left(\tfrac{\text{ft}}{\text{s}}\right)\,\!</math>	<math>E_k^{} \left(\tfrac{\text{mJ}}{\text{mm}^2}\right)\,\!</math>
3B (k26 and perfect seal)	175	9.0
SNAP 5 with hopper	190	11
SNAP 2 with hopper	200	12
PAC with hopper	200	12
Ryan's singled +bow	210	13
The U3's SM 1500	250	18
Ice9's Big Blast	350	36

KED limits

A critical KED (i.e. one that will start to damage a surface) is given the notation <math>E_{crit}^{}\,\!</math>.

The values for eye, skin, and bone damage below should be thought of as good estimates of the average values. Much more testing is necessary to determine the average value and distribution of the KED necessary to penetrate the body. This testing is gruesome, as it generally involves shooting dead rabbits or pigs in the eyes, and future testing is not likely to be performed for this reason. Please use generous safety factors. This information is provided to put the power of a Nerf gun into perspective so that this power may be fully respected.

A note for the squeamish

Most of the references containing critical KED data also have photographs of gunshot wounds that some readers may find to be disturbing. Don't look at these references if you are squeamish.

Hasbro (SRS-045)

Hasbro's SRS-045 sets a KED limit of 1.6 <math>\tfrac{\text{mJ}}{\text{mm}^2}</math> for projectiles with a KE over 80 mJ. This limitation is based on ISO #8124. Hasbro usually designs for 20% under this limit to account for manufacturing variations.

Eye damage

Some data for spheres is available. Admittedly, spheres are not particularly similar to Nerf darts, but many darts have domed heads, so the data should be correct in that case, and approximately correct for flat headed darts.

For a summary and analysis of data available up until 1994, see Sellier and Kneubuehl.

This data suggests there is a relationship between critical KED and projectile diameter. However, the data for diameters 3.2 mm and higher (most Nerf darts have a diameter of about 12.7 mm) is fairly consistent, and that data has an average value of <math>E_{crit}^{}\,\!</math> of 62.0 <math>\tfrac{\text{mJ}}{\text{mm}^2}\,\!</math>.

This data is for penetrating the eye, not simply causing damage. The eye can be damaged at levels below the listed critical KEDs.

<math>d \left(\text{mm}\right)\,\!</math>	<math>m \left(\text{g}\right)\,\!</math>	<math>V_{tsh} \left(\tfrac{\text{m}}{\text{s}}\right)\,\!</math>	<math>E_{crit}^{} \left(\tfrac{\text{mJ}}{\text{mm}^2}\right)\,\!</math>	Source
1	0.004	70	12.5
2.36	0.054	71	31.1
3.2	0.135	101	85.6
4.34	0.356	75	67.6
4.37	0.341	75	64.0
4.37	0.513	58	57.5
6.4	1.037	47	35.6

Skin damage

An analysis of multiple sources suggests an average critical KED value of about 175 <math>\tfrac{\text{mJ}}{\text{mm}^2}\,\!</math>.

These values are for skin penetration (i.e. breaking or becoming embedded in the skin) not bruising. Bruising likely occurs are far lower KED values.

<math>d \left(\text{mm}\right)\,\!</math>	<math>m \left(\text{g}\right)\,\!</math>	<math>V_{tsh} \left(\tfrac{\text{m}}{\text{s}}\right)\,\!</math>	<math>E_{crit}^{} \left(\tfrac{\text{mJ}}{\text{mm}^2}\right)\,\!</math>	Source	Notes
4.50	0.535	101	171
5.56	1.07	74.7	122
9.65	7.32	58.2	170
4.50	0.51	117	221
11.25	8.5	70	210
8.5	4.5	71.3	202
4.5	0.54	110.4	207		lead spheres
4	0.31	120.6	179		brass spheres
4	0.26	126.1	164		steel spheres
4	0.08	197.8	125		glass spheres
4	0.05		85		plastic spheres

Bone damage

Data summarized by Sellier and Kneubuehl leads to an average critical KED value of about 200 <math>\tfrac{\text{mJ}}{\text{mm}^2}\,\!</math>.

Damage to material (linear elastic case)

For more information about stress and strain see the strength of materials page.

The material can be assumed to deformed in a simple manner as a simplification: the projection of the impact area deforms like a linear spring and nothing else deforms.

How much energy per unit volume can a material absorb before permanently deforming? This energy is called <math>U_r\,\!</math>, the modulus of resilence. The answer can be found rather easily with Hooke's law and the work-energy theorem (normalized per unit volume). <math>\sigma_y\,\!</math> is the yield stress and <math>\varepsilon_y\,\!</math> is the yield strain,

<math>W = \int F dx \longrightarrow \underbrace{\frac{W}{L^3}}_{W^{}} = \int \underbrace{\frac{F}{L^2}}_{\sigma} \underbrace{\frac{dx}{L}}_{d \varepsilon} \longrightarrow W^{} = \int \sigma d \varepsilon\,\!</math>

<math>\sigma = E \varepsilon\,\!</math>

<math>U_r = \int_0^{\varepsilon_y} \sigma d \varepsilon\,\! = \int_0^{\tfrac{\sigma_y}{E}} E \varepsilon d \varepsilon = \left[\frac{E \varepsilon^2}{2}\right]_0^{\tfrac{\sigma_y}{E}} = \frac{\sigma_y^2}{2 E}</math>

This material property can be multiplied by the thickness of the material to find a critical KED value for that material for yielding.

<math>E_{crit}^{} = U_r t = \frac{t \sigma_y^2}{2 E}\,\!</math>

Note that yielding really only is denting; if the KED value is less than the critical KED value, the material will completely defeat the projectile without any denting (unless fatigue is considered). This information can be directly used to determine whether eyewear will protect against a dart.

If the KED value is greater than the critical KED value, all that can be said is that at least the material will be slightly dented. Potentially the projectile could dent or fracture the target and bounce off. Potentially the projectile could be embedded in the target. Potentially the projectile could pass completely through the sheet. This analysis offers little indication about which is more likely. Sellier and Kneubuehl have some information about projectiles moving through objects they impact.

Different critical KED values for complete defeat of the material must be found experimentally or via finite element analysis.

Measuring KED

Chronograph, scale, and calipers

A chronograph, a small scale, and potentially some calipers can be used to find KED. Measure the muzzle velocity of the blaster, the mass of the dart, and the diameter of the dart. These numbers can be directly plugged into the KED equation.

Ballistic pendulum

A ballistic pendulum can be used to directly find the kinetic energy of the dart, and from there the KED can be found by dividing the KE by the minimum area. A ballistic pendulum is far easier to build than a homemade chronograph, as it only involves a pendulum and some weighted target for the projectile. A styrofoam target should work nicely for Nerf applications.

Wikipedia has more information about ballistic pendulums.

Material penetration

Using the information in the "Damage to material" section above, one can figure out what thickness of a certain material is necessary to dent it. Practically speaking, however, this is not useful as denting any material a Nerf gun could easily shoot through like cardboard is very simple.

Empirical testing with a sheet of cardboard or something similar can find the KED necessary to penetrate the sheet, and this sheet can then be used to test whether the KED of a blaster is below or above a certain limit.

Other notations

Sellier and Kneubuehl use <math>E_{ths}^'\,\!</math> for the critical kinetic energy density. "Ths" refers to a threshold; this is the threshold to start penetrating something. This notation's use of a prime does not make sense as the primes in this context generally denote normalization with respect to one length. Two primes are necessary for normalization with respect to an area (i.e. a length squared).