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itle{fseries Study of maximum exit velocity and the motion inside the blowpipe}egin{abstract}This analysis applies both theoretical deduction and its subsequent mathematical simulation and the experimental verification to the motion of projectile in the blowpipe. In the theoretical part, the deduction is based on aerodynamics and classical mechanics.

While in the experiment part, the effect of different length of the blowpipe has been studied and the motion of the projectile is recorded and quantified. Comparisons with different pipe lengths between the theoretical prediction and the experimental data are then accomplished and showed in figures.end{abstract}egin{keyword}blowpipe, aerodynamicsend{keyword}end{frontmatter}egin{figure}Hlabel{fig1} centeringincludegraphicsheight=2.5in{man.jpg}caption{An aborigine using a blowpipe as weapon.cite{8} }end{figure}section{Introduction}Recently, I watched a movie called emph{Pirates of the Caribbean}, in which the main character was washed to an island, where many aboriginal people lived there. They used blowguns, also called  blowpipe, to make the protagonist comatose.

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At that time, I was really intrigued by this amazing little device. How can our mouth be able to generate such a high velocity that can attack human beings in such a long distance?According to Merriam Webster, Blowpipe is defined as “a tube through which a projectile (such as a dart) may be impelled by the force of the breath” cite{1}. According to James Stuart Koch, “The history of blowgun could date back to the stone age and was known and used at that time on all the inhabited continents, with the exception of Africa” cite{2}. Usually, the projectile used in the blowpipes are pellets or darts. While pellets seem to be the projectile of choice for birds and the smallest game, darts could be used for much larger targets and were sometimes poisoned cite{2}. In my experiment, due to safety issue, I change the projectile into light-density paper clip because even if it accidentally shoots towards any person, the light-density projectile always has lower momentum, thus decreasing the harm. I also put a wide-spread sponge cushion in the front so that the projectile can land on a soft-made ground with little derogation.Blowgun now is still used as a useful weapon in Amazon tribes since it is as fast and as accurate as shooting.

My passion towards this fascinating weapon is that I really want to know how can a small projectile become such a pernicious killing weapon. I want to know how it works and under given condition, how large the momentum it could generate. In order to narrow my data quantity, and by searching online for proper length and diameter of the real blowpipe, I finally come to determine that the tube length is between 1.0m to 1.8m while the diameter is determined to be 1.6cm. I want to investigate the influence of the length of tube on the maximum exit velocity and moving motion, given the fixed diameter of the tube.Research Question: What is the maximum exit velocity of the projectile and the motion of it inside the blowpipe as a function of time, provided that the blowpipe’s length is between $1.

0 m $ to $1.8 m$ and the diameter $1.6 cm$?My teacher encouraged me to use high-speed camera and computer modeling to aid my experiment. The purpose of my physics exploration is two-fold. First, I set up the equipment by using acrylic tube bought online, paper-clip material, and respirometer; the tube was set on the shelf, with a projectile set inside, and one of friends with great lung capacity blew the pipe hard.The second part of my experiment is the analysis of the data I acquired. By referring to the equation, the Matlab can draw experimental graph and theoretical graph, making it convenient for me to evaluate and analyze.

section{Theory-based deduction}To accomplish my goal, the projectile is made by the paper clay, with a density of 0.25$~g/cm^3$; and the gas is considered as an ideal gas. First, we have to get the equation of the whole experiment. The first step is to determine the shape of the projectile. Any object moving in the air will have a resistance force and a drag force: The drag coefficient is defined as a dimensionless quantity that is used to quantify the drag force of an object in a fluid environment, such as air or water.The drag force equation is given by cite{4}:egin{equation}label{eqn1}F_d=frac{1}{2}ho u^2 C_d A,end{equation}where $ho$ is the density of the object, u the velocity of the object, $C_d$ the drag coefficient, and A the projected area. To have small drag coefficient for higher speed, the drag coefficient should be relatively small.

emph{Cone-shape} ($C_d=0.20$), emph{streamlined body} ($C_d=0.04$), and emph{streamlined half-body} ($C_d=0.09$) have ideal drag coefficient and they all have a streamlined body shape cite{4}. However, the major driving force for the projectile to move in this experiment is the pressure difference produced during the blowing process. And the force given here equals to the product of pressure and the stressed area. Also, compared with the drag force, this driving force is much more important, indicating that we need a large projected area.

 Therefore, a body with a small drag coefficient does not mean that it is an ideal one. Due to the fact that the pressure force is the most important force in the whole experiment, the stress area should be big enough to make this source of force outweigh the others.Taking all the factors into consideration, the streamline-shaped cone is the best choice. Thus, a free-body diagram can be drawn:egin{figure}Hcenteringincludegraphicsheight=2in{skematic.pdf}caption{This figure illustrate the quantity we will employ.}label{fig2}end{figure}Based on the diagram above, and by referring to aerodynamics, we have the following force equation:egin{equation}label{eqn2}(P_1-P_infty) S_1-f-F_d+I=Mcdot frac{du}{dt},end{equation}where the first term comes from the pressure difference obviously, $f$ is the air resistance (this term is negligible compared to the others), $F_d$ has been given above, and $I$ is the instantaneous blowing force, namely, the collision force between gas particles and the projectile.

The whole theoretical equation is based on Newton’s Second law: $F=ma$,where $M$ is the mass of the projectile; $du/dt$ is the acceleration of the moving projectile. Knowing the theoretical equation, we should now demonstrate how to deduce the elements in the equation. Let us start from the state function of ideal gas:egin{equation}label{0}P=frac{ho RT}{M_r}=ho R_g T,end{equation}where $ho$ and $M_r$ are the density and Molar mass of air, respectively; $R_g=R/M_r;$and here,$$m=m_0+int frac{dm_1}{dt}dt-int frac{dm_2}{dt} dt,$$where $m_0$ is the original gas mass in the left area, $m_1$ represents the blowing mass and $m_2$ is the mass leaking through the gap between the projectile and the tube.The rate of leaking can be represented by$$ frac{dm_2}{dt}=ho  S_{gap} v_{rel}=ho (S_{tube}-S_1)(v_2-U),$$where $v_{rel}$ is the relative velocity, $v_2$ represents the fluid speed at the gap between the tube and the bullet, and $U$ stands for the projectile’s velocity.

Finally, we employ emph{Bernoulli’s Equation}cite{5}:egin{equation}label{eqn3}P_1+frac{1}{2}ho v_1^2=P_infty+frac{1}{2}ho v_2^2,end{equation}where $P_1$ is the air pressure to the left of the projectile, $P_infty$ the air pressure at  the gap of the leaking place, $v_1$ the velocity of the fluid at the left side of the projectile and $v_2$ the same one as above (also called the leaking rate).Then we haveegin{equation}label{eqn4}v_2=sqrt{frac{P_1-Pinfty}{ho}cdot2+v_1^2}.end{equation}We must be cautious that Bernoulli’s Equation is only valid for incopressible flows given that the fluid’s velocity is lower than $0.3  Mach$.

After examining the gas, we can find the speed of gas is far lower than the threshold, confirming the validity of this equation.With $v_2$ obtained, all the other variables can be displayed, thus can be put into emph{Matlab}. (See the Appendix at the end.)section{Experimental Set-up}Materials needed:egin{itemize}itemA. 5- the  acrylic tube of 1.

0m, 1.2m, 1.4m, 1.6m, and 1.8m;itemB.

1- paper-clay material;itemC. 1- high speed camara;itemD. 1- a computer;itemE. 1- a respirometer;itemF. 1- a balance.

end{itemize}egin{figure}Hcenteringincludegraphicsheight=2.5in{equipments.jpg}caption{All the necessary apparatus.}label{fig3}end{figure}The advantage of utilizing a high-speed camera is obvious: We can get the instantaneous speed, and accurate maximum exit speed along with a software Tracker; we can use the camera to record the whole process and analyze it since the tube is transparent.egin{itemize}item1. Set up the tube on the supporting equipment;item2.

Calibrate the Gradienter to make sure that the projectile’s motion will not be influenced by its weight;item3. Use the respirometer to measure the rate of blowing of the person;item4. Use the balance to measure the mass of the projectile;item5. Blow the pipe, with a pressure meter measuring the pressure in the tube;item6. Use the high-speed camera to record the motion in the form of video;item7. Use the tracker to measure the experimental data;item8.

Use Matlab to give the theoretical data.end{itemize}egin{figure}Hcenteringincludegraphicsheight=2.5in{apparatus.jpg}caption{Assembled apparatus.

}label{fig4}end{figure}      egin{table}H      centering egin{tabular}{|p{3.6cm}|p{3.6cm}|p{4cm}|}      hline      Dependent Variables (what are measured) Variables (what are changing) Variables (what are controlled) hline      The maximum exit velocity length of the tube diameter of the tube   hline       The air pressure when blowing&   & The rate of blowing hline                                                 &   bottom surface area of the projectile hline      & mass of the projectitle hline      end{tabular}       caption {The assortments of variables in this experiment.

}      end{table}            egin{table}Hcentering egin{tabular}{|p{6cm}|p{4cm}|}      hline     Blowing rate&$6.73 imes 10^{-3} m/s$ hline      Diameter of the bottom of the projectile&$2.5 cm$ hline       Mass of the projectile&$0.

11 g$ hline      Density of the projectile&$0.25 g/cm^3$ hline      Drag coefficient&$0.20$ hline      end{tabular}       caption {The magnitudes from emph{DATABASE}.

}      end{table}%—————————————————————————section{Results and Analysis}      egin{table}H      centering egin{tabular}{|c | c |c |}      hline       Length of the tubes (m)&Theoretical value (m/s)&Experimental value (m/s) hline      1.0&28.71&33.

63 hline       1.2&28.51&32.10 hline       1.4 &  27.

99 &32.50 hline      1.6& 27.39& 32.10 hline      1.

8&26.76&31.89 hline      end{tabular}       caption {The data of maximum exit velocity for different tube lengths (diameter=$1.6 cm$).}      end{table}Margin of error is not presented because it was not provided in the databaseThe information is presented in a graph below. In the following figures blue lines represent the theoretical data, while the red lines are the experimental data. egin{figure}Hcentering%???????????????subfloatLength=1.0 m{includegraphicswidth=2.

5in{0m.pdf}label{10}}subfloatLength=1.2 m{includegraphicswidth=2.5in{2m.pdf}label{12}}end{figure}egin{figure}HcenteringsubfloatLength=1.

4 m{includegraphicswidth=2.5in{4m.pdf}label{14}}subfloatLength=1.6 m{includegraphicswidth=2.

5in{6m.pdf}label{16}}end{figure}egin{figure}HcenteringsubfloatLength=1.8 m{includegraphicswidth=2.5in{8m.pdf}label{18}}subfloatLength=1.8 m{includegraphicswidth=2.5in{x.pdf}label{x}}caption{The v-t plots for different blowpipe lengths and x-t plot for the length of 1.

8m. All the legends have been labeled in figures.}label{fig5}end{figure}%——————————————————————————————-section{Data Analysis}When the length of the tube is changed, the maximum exit velocity changes at the same time. According to the the graph and data above, the projectile reached its maximum exit velocity at $1.0 m$, $1.6 cm$ tube, and the theoretical value and real value are $28.7 m/s$ and $33.6 m/s$, respectively.

According to the velocity-time graph, we can find that when the tube is over $1.0  m$ long, its maximum velocity is always reached in the middle of the process, but then the projectile’s speed begins to decrease, thus making the exit speed smaller than the optimum value.We can also find that for a tube with $1.6 cm$ of diameter and given the blowing rate and atmospheric pressure, the optimum length of the tube with the biggest projectile exit velocity is $1.0 m$.

Generally, my experiment  is roughly correct since the trends of all my experimental graphs look really similar to the theoretical graphs. Besides, the exit time points are nearly the same for every group of data. And further, the $x-t$ plot of prediction (Fig. ef{x}) agrees quite well with that of the experiment.section{Error Analysis}A weakness in the investigation is the over-simplification of the physical scenario and the corresponding over-idealized model here. From the theoretical graphs above, it is not hard to deduce  that, in the ideal situation, the motion and the speed of the projectile at different positions of tubes are the same. In other words, the projectile would always reach its maximum velocity at one-meter position.

However, this will not happen in real-life situation because when the projectile moves closer to the open end, the leaking rate will get smaller, making the air pressure at the left side higher than our expectation, and thus higher the exit velocity. Hence, not only will not the projectile reaches maximum velocity at one-meter position all the time, but the projectile’s exit velocity would be faster than the theoretical value. That is the main source of the discrepancy between the ideal data and experimental data in the table. As for our deduction, first, the description of gas by Eq.ef{0} is not quite accurate since it is for the ideal gas and not for the air. Second, the other fundamental description (Eq.

ef{eqn1}) is also a experiential one rather than a accurate theoretical deduction. Third, as the bullet approaches the end, the Eq.ef{eqn3} will give a poor description of its motion since it’s based on quasi equilibrium assumption. The only way to make a accurate prediction is to base the whole deduction completely on the formidable emph{Navier-Stokes equation}, a field equation frequently applied in airplane-design and famous for its solving difficulty.

However, according to the trend of the graphs, those errors are still acceptable, indicating the overall validity of the analysis.The maximum exit velocity obtained is $28.7 m/s$, while the theoretical value is $33.6 m/s$. The result is not accurate enough. Here is the percentage accuracy error:egin{align}\%Error&=frac{Measured-Expected}{Expected} imes100\%  onumber             &=frac{33.

6-28.7}{28.7} imes 100\% onumber             &=17\%. onumberend{align}%—————————————————————————section{Motion of the projectile in the tube}The motion of the projectile is similar to the form as shown in Fig.ef{fig5}   below since the surface of the projectile is not flat.

According to emph{Bernoulli’s principle} cite{6}: In fluid dynamics,  an increase in the speed of a fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid’s potential energy.egin{figure}Hcenteringincludegraphicswidth=3in{motion.pdf}caption{The schematic diagram for the motion of the projectile.}label{fig5}end{figure}egin{figure}HcenteringsubfloatThe pitching-up attitude.{includegraphicswidth=2.2in{A.

pdf}label{a}}subfloatThe pitching-down attitude.{includegraphicswidth=2.2in{B.pdf}label{b}}caption{The the schematic diagram for the attitude of the projectile.}label{fig6}end{figure}%?????????????????????????????????????????Thus, just like a plane, as shown in Fig ef{a} and Fig ef{b}, when the airflow passes the irregular shaped cone, the upper pressure would be smaller than the lower pressure, causing the projectile to pitch up; similarly, when the projectile reaches the upper floor, it will pitch down.

In this case, we would obtain a pitching moment motion.%————————————————————————————-section{Conclusion and Evaluation}To answer my research question, the maximum exit velocity of the blowpipe with a diameter of 1.6cm is $33.6 m/s$, with a length of $1.

0 m$. The motion of the projectile inside the tube is a pitching moment track, as shown in  Fig ef{fig5}. A strength of this investigation is the use of mathematical modeling via Matlab, on which the experimental data and the theoretical data are compared. Besides, the use of high-speed camera and the App Tracker also provide me with an accurate demonstration of the experimental data. Besides, there was no safety issue during the whole process, since we have set up the cotton protective equipment in the front, and there was no other people in the room except me and my friend.This investigation can be applied to  other fields of study, such as the motion of a bullet inside a gun and the maximum velocity it can reach, thus deducing the optimum length of the gun tube. This would be useful to the weapon production industry. Besides, this investigation can also be utilized in measuring the projectile velocity of an air canon, which is a device similar to the blowpipe cite{7}.

 Future study can be done to investigate the effect of different diameters of the tube.

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