# Archive for the ‘Exterior ballistics’ Category

## Nerf Engineering notes

Posted by btrettel on July 8, 2012

I’ve started organizing my writings and thoughts on Nerf ballistics into a set of notes (link corrected). Readers of my blog might be interested.

For the moment my notes are very rough and incomplete. I’ll update the PDF as I make changes. I’d appreciate any feedback, especially corrections and ideas.

Edit (2012-10-02): I’ve removed the notes because they need a lot of work before I’m comfortable distributing them.

Edit (2013-01-19): I’ve readded the notes as I’ve revised them somewhat. They are still a work in progress.

## Solving the point mass model of a dartâ€™s trajectory, part two

Posted by btrettel on June 12, 2011

I wrote a technical paper about solving these equations and made some conclusions for design. I have fully solved the flat-fire case given that the x velocity stays above a certain number (and it generally will).

The take-home message is pretty simple: Keep kinetic energy constant. If you reduce the drag coefficient multiplied by the cross-sectional area, you’ll increase range. If you reduce the muzzle velocity divided by the dart mass, you’ll increase range.

The latter conclusion is not too surprising to me as I already knew dart mass had an effect like that, but to quantify the effect is very nice.

I’d happily accept corrections, comments, questions, suggestions, and whatever else.

Posted in Exterior ballistics, Math | 2 Comments »

## Solving the point mass model of a dart’s trajectory, part one

Posted by btrettel on June 7, 2011

This blog post is relatively math heavy and it serves mostly as notes for my own use. I will assume some familiarity with differential calculus and ordinary differential equations. I also will assume some familiarity with the basic equations that govern the motion of Nerf darts after they leave the barrel.

Yesterday I had an epiphany while in the shower about a potential approach to solve the equations for the trajectory of a point mass with quadratic drag. This is a simplification to the standard equations of motion for the trajectory of a point mass in 2D with quadratic drag. These equations are presented below without derivation (or an explanation of what $k$ is).

EOM in x: $\ddot{x} = -k \dot{x} \sqrt{\dot{x}^2 + \dot{y}^2}$

EOM in y: $\ddot{y} = -k \dot{y} \sqrt{\dot{x}^2 + \dot{y}^2} - g$

These are two coupled non-linear ODEs. To the best of my knowledge, there is no known solution in terms of elementary functions. You either have to define new functions to solve these equations or solve them numerically, as I have done in the past.

But we can do a few tricks to get an okay approximation. Note that the vast majority of shots from Nerf guns are level. Basically, this means that the y component of velocity is far smaller than the x component. If we ignore drag and solve $\ddot{y} = -g$ directly, we can find that the maximum velocity of a Nerf dart in the y direction is less than 10 m/s. But Nerf darts almost always will be traveling faster than 30 m/s. If we assume that $\sqrt{\dot{x}^2 + \dot{y}^2} \approx \dot{x}$ we commit at worst a 5% error if the x velocity stays above 30 m/s. This is the assumption I will make.

For an even better approximation, there are a few things that could be done. A perturbation method could be used. A guess for $\dot{y}$ could be made; $\dot{y} = -g t$, the y velocity neglecting drag, might be good for a conservative approximation. We could assume that $\dot{y}$ is a function of $\dot{x}$ or another variable (like the height, or range), perhaps even a multiple of it. The multiple case reduces to an extra factor the drag coefficient (already a fudge factor for our purposes) can absorb. I will not take any of these approaches as I am only exploring potentially valid solutions now.

With my assumption in mind, I’ll rewrite the EOM.

EOM in x: $\ddot{x} = -k \dot{x}^2$

EOM in y: $\ddot{y} = -k \dot{x} \dot{y} - g$

These equations are now uncoupled and only the first one is non-linear. I can solve for $\dot{x}$ directly and substitute that into the second EOM.

$\dot{x}$ can be solved for by doing the following change of variables:

$z \equiv \dot{x}^2 \longrightarrow \dot{z} = 2 \dot{x} \ddot{x} \longrightarrow \ddot{x} = \displaystyle \frac{\dot{z}}{2 \sqrt{z}} \longrightarrow \dot{z} = -2 k z^{\tfrac{3}{2}} \longrightarrow z^{-\tfrac{1}{2}} = k t + c$

I can substitute back in my definition for z to get $\dot{x}$. I’ll use the initial condition $\dot{x}(t = 0) = V_0$ to get the complete solution.

$\dot{x} = \displaystyle \frac{1}{k t +c} \longrightarrow \dot{x}(t) = \displaystyle \frac{V_0}{V_0 k t + 1}$

Now we must solve the EOM in y. The equation is written below with $\dot{x}$ substituted in. I defined $A \equiv k V_0$.

$\ddot{y} = -\displaystyle \frac{A \dot{y}}{A t + 1} - g$

I will use a change of variables and omit most of the details. $\xi = \text{ln}(A t + 1)$

The ODE with these changes becomes $y_{\xi \xi} = - \displaystyle \frac{g e^{2 \xi}}{A^2}$ where the subscript represents differentiation with respect to that variable. The general solution for this ODE is $y = B \xi + C - \displaystyle \frac{g e^{2 \xi}}{4 A^2}$.

Substituting back in for y and t and using the ICs $\dot{y}(0) = 0$ (the dart does not start with any vertical velocity as it is fired flat) and $y(0) = h$ where h is the height of the barrel I arrive at the following equation for the dart’s y position as a function of time.

$y(t) = h + \displaystyle \frac{g}{2 (k V_0)^2} \text{ln}(k V_0 t + 1) - \displaystyle \frac{g t}{2 k V_0} - \displaystyle \frac{g t^2}{4}$

This is correct. You can solve for $\dot{y}$ and $\ddot{y}$ and substitute back into the ODE to check. You can also check the ICs; they are satisfied. Additionally, the solution approaches the solution without drag. At first I thought it did not, but it can be shown to approach this solution exactly with Taylor series expansion.

To get the range, we must find timeÂ  the dart hits the ground. Yet the equation above can not be solved for t explicitly.

So I can find an approximate solution that is undesirable because I can’t proceed to get the range equation. I can think of a few options from here. One is to approximate the equation for y to find an approximate time. Another is to modify the original ODEs in a way that allows for the time to be found. Yet another is to use the identity $\ddot{x} dx =\dot{x} d \dot{x}$ (and the equivalent for y) to avoid time all together, but this approach has another problem: what $\dot{x}$ and $\dot{y}$ correspond to when the dart strikes the ground?

In my next post I will prove that this solution is correct and converges to the solution without drag as $k \rightarrow \infty$ and develop some approximations for different scenarios.

(I had some incorrect ramblings here before. They have been removed.)

Edit: In the next post I will show how the form of the solution for $x(t)$ probably will save us and allow an explicit equation for the range to be formed. $x(t) = \displaystyle \frac{\text{ln}(V_0 k t + 1)}{k}$. Note that this allows us to replace the ln term in the y equation with a multiple of x. x = R (the range) when the dart hits the ground… this may allow us to find the time the dart hits the ground as a function of R, which we can substitute back into the x equation to solve (hopefully) for R. That is the outline of what I will attempt now.

Edit again: I believe this is an adequate approximate equation for the range of a flat fired Nerf gun:

$R = \displaystyle \frac{1}{C} \text{ln} \left(\frac{h (C V_0)^2}{g} + 1 + \text{ln} \left(\frac{h (C V_0)^2}{g} + 1 \right) \right)$, where $C \equiv \displaystyle \frac{\rho_{atm} C_d A}{m}$ .

An okay simpler approximation follows.

$R = \displaystyle \frac{1}{C} \text{ln} \left(\frac{h (C V_0)^2}{g} + \frac{27}{25} \right)$

## My challenge to Nerf rifling believers

Posted by btrettel on April 17, 2011

Do you believe most Nerf darts can be made more stable by spinning? If so, this challenge is for you!

The challenge is simple: Provide objective and statistically significant evidence that spinning darts improves stability, precision, or range.

Some notes

Rifling barrels for Nerf has already been thoroughly debunked. To say it in brief, the vast majority of Nerf darts are already stable, so theyÂ  have nothing to gain by spinning. In fact, they have a lot to lose, and this is often ignored by rifling believers.

So far, rifling believers have singularly failed to provide strong evidence to support their claims. Whenever they do, their results are shown to actually not be statistically different than what you would expect from smoothbores.

Also, tests are very often completely subjective. The results of many tests are simply “it looked better.” A subjective test like examining the dart’s trajectory could potentially be useful, but only when the experiment is properly designed. The tests must be blinded to have any shred of objectivity.

I don’t care how the spinning is done as long as the darts can be verified to be spinning too.

## Rifling: Helpful, harmful, or ineffective?

Posted by btrettel on January 7, 2011

By making analogies with real guns, some Nerfers have proposed that rifled barrels may be beneficial for Nerf blasters. But is this true? I will examine the two most popular claimed benefits of rifling, that rifling increases range and improves accuracy, and conclude that rifling as implemented thus far has had no significant effect on range or accuracy and it is not likely to have any effect under any circumstances.

First, the reader must realize that these claims are made most often without any backing. The hypothesis that rifling improves accuracy or range is often made based on misunderstandings of what rifling does. Spinning projectiles do not have less drag. Projectiles are spun to improve stability, as I will explain.

Stability of projectiles

A projectile is stable if it flights straight without overturning. This is desirable as the overturning motion reduces accuracy and range.

Rifling is used to improve the stability of a projectile’s flight. But can the stability of a Nerf dart be improved? In general, the answer is no because Nerf darts get their stability from static rather than dynamic characteristics of the dart.

The simplest way to make a stable projectile is to put the center of gravity far in front of the center of pressure. Details as to why this is stable will be later written in the Wiki. Most Nerf darts get their stability in this way; this is why darts are weighted at their nose.

But, most real bullets are made of a single material and they do not have this desirable weight distribution. Spinning the bullet around its longitudinal axis (as rifling does) can stabilize bullets in this case.

So, by simple examination of the mechanisms involved, we can conclude that rifling won’t have any significant effect on darts with the right weight distribution. Those darts are already very aerodynamically stable. There is no reason to rifle as there will not be any real benefit.

Some benefit from rifling seems plausible for very light darts that do not have the right weight distribution. But this is not an argument for rifling necessarily; adding weight to the front is by far the easiest way to stabilize these projectiles. However, this may not seem to be an acceptable choice for some Nerfers. Very lightweight darts may be desirable for safety reasons, however, there are other ways to improve safety of a dart (like reducing the muzzle velocity) that are far simpler than rifling.

There are many potentially significant disadvantages to rifling that most proponents of the idea are unaware of. I detail the disadvantages that come to mind below.

• Increased friction – If done poorly, the rifling could increase friction in the barrel and potentially reduce performance as a consequence.
• Leaks around projectile – If done poorly, the rifling grooves could allow for air to leak around the projectile, reducing performance.
• Increased complexity of building – Smoothbore barrels are simpler.
• Less translational KE – To have a spinning dart, some of the energy that would have been put into translational kinetic energy and have contributed to range is instead put into rotational kinetic energy. Rifling is beneficial when this trade-off improves stability such that range or accuracy is improved satisfactorily. However, the reduction in translational KE may not be acceptable in all cases.
• Reduction of stability – Poorly made darts may not have their weight distributed evenly around the longitudinal axis of the dart. Spinning could destabilize these darts and reduce range and accuracy.

Examining the accuracy claim with data

In 2009, a Nerfer who went by the handle Landru did some tests to see what effect spinning a dart had on accuracy. He used a setup with a spinning barrel. It is believed that this spinning barrel provides a way to control the spinning without making multiple rifled barrels. The test did not address rifling directly, rather, it addressed the question of whether spinning darts could even improve accuracy.

Landru posted some data that he claimed showed that the standard deviation of the locations of darts spun at 2000 RPM was lower than that from no spinning.

However, Landru neglected any sort of statistical analysis. I made a brief post that demonstrated his methods were flawed. I used an f-test to see whether there was any statistically significant difference between the two groups. Assuming a sample size of 20, I found critical f-values of 0.46 and 2.12 for$\alpha$ = 10%. The f-value of was 1.49. As this was between the critical values, the differences were not statistically significant and consequently we can not determine if they were due to the rifling or random chance.

Examining the range claim with data

Back in perhaps 2003 or 2004, a Nerfer who went by the handle Vassili tested rifled PETG barrels. He found that the average range of rifled PETG was higher than that of smoothbore PETG. Thankfully, Vassili didn’t claim rifling improved range directly. He only offered a tautology: “When it worked, it worked.” But did it work? Can we attribute any of the differences to the rifling and not random chance?

No, we can not. A t-test suggests the two data sets are statistically the same at the $\alpha$ = 5% level. The critical t-value is 2.65. The t-value of the test for the mean is 1.21. As this is within the bounds we would expect at the 5% level of error, we can confidently state that rifling did not increase range in this case.

However, it can be shown that rifling increases the standard deviation of the range with an f-test (data to be added later). This should lead to a decrease in precision due to a decrease in repeatability (each shot is more variable). It also shows that more shots will have lower range with rifling. These two disadvantages are significant.

Conclusion

Based on the implausibility of the explanation for the benefit for rifling and the lack of evidence to suggest that rifling provides any benefit for Nerf darts, I conclude that rifling is ineffective at best and harmful at worst for Nerf.

2014-09-05: Comments disabled due to spam.