Accuracy of modelling projectiles using classical

mechanics

Kyle Byrne

Abstract

This report will be an effort into finding the accuracy available inmodelling the paths and behaviour of prejectiles using classical mechanics.I hope to achieve this by carrying out a set of projectile experiments,filming and then from this extracting the parabola formed by the path ofthe projectile. I will then compare this to the predicted parabola as foundby classical mechanics, described below. By comparing these paths I willhave a visual representation of the affect air resistance and other externalforces have on the projectile.Through the values I record I will also beable to extract numerical values for characteristics of the flight which Ican compare to my real world flights as well. The motivation behind thissubject is to find out just how good of an approximation we can find whenleaving out values such as air resistance.

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1 Background Physics

To model a projectile with initial velocity = u fired from an angle θ to thehorizontal we must first split it into its horizontal and vertical components.Doing so can give uvertical = u sin θ anduhorizontal = u cos θ. From these we can

simply substitute into s = ut+ at2

2 to find the horizontal and vertical distances.

svertical = y = u sin θt− gt2

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shorizontal = x = u cos θt as ahorizontal is 0

Rearranging the horizontal component we can find that t = xu cos θ and substi-

tuting this into the vertical we can find

y = u sin θx

u cos θ−g xu cos θ

2

2

y = x tan θ −g x2

u2 cos2 θ

2

y = x tan θ − gx2

2u2 cos2 θ

y = x tan θ − gx2 sec2 θ

2u2

This result will provide the basis for the visual comparison side of this reportby allowing us to predict what path the projectile should take.We can derive the time of flight and horizontal range by using the equationy = ut+ 1

2at2 using the vertical component of velocity we get

y = u sin θ − 1

2gt2

And so

t(u sin θ − 1

2gt) = 0

From this we can see that

t = 0 or t =2u sin θ

g

this time value is known as our time of flight , it is the time taken to fall backto an initial height after projection.

To find the x co-ordinate at this time and hence the range the particle travelswe can this time value into the equation S = uhorizontalt we then find that

Srange = u cos θ t

Srange =u cos θ 2u sin θ

g

Srange =2u2 sin θ cos θ

g

Srange =u2 sin 2θ

gas 2 sin θ cos θ = sin 2θ

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With these values I will be able to both visually and numerically compare thepredicted flight with the real flight. Values for the real flight will be found bytracking the particle through its flight at correct scale, this will give us actualx , y ,t and S values for any time with our recorded velocity and angle.

2 Preliminary trials and research

Through initial testing it became clear that there was a number of problemswith the experiment , specifically ;

• Method of projection

• Distance measuring

• Velocity measuring

2.1 Method of Projection

The first problem I came across was how I would actually fire the projectile, Ishortlisted 4 different methods.

• Elastic bands

• Sling (e.g. trebuchet)

• Air powered

• Spring

By assessing the characteristics I was looking for I could rule out 3 of these meth-ods. Elastic bands would be too unreliable, not providing consistent repeatablepower. The sling method would add rotation to the projectile, meaning it wouldinteract with the air differently and so any progression into calculating the effectof air resistance would prove far more difficult. The air powered method whilstprobably being the most repeatable would have the problem of being probablytoo powerful, being difficult to carry out the low power testings I hoped to do,also this method would be the hardest to get hold of the apparatus for.So this leaves Spring powered, I managed to get hold of a spring powered cannonwhich provided suitable, repeatable and changeable power output, and also the

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added bonus of easy aiming and angle measurement.

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2.2 Distance Measuring

The premise of my experiment was about comparing visually and so for thisit became quite apparent quite soon that I would need high speed video to beable to actually extract any sort of data from the footage. The problem withthis however is that as you increase FPS you decrease in resolution and so thismeant that my original plan of having metre rulers in shot for measuring wouldno longer work, as they weren’t readable. The solution I came up with wasusing a striped tape measure with width of 4cm not for measuring but insteadjust to use as a scale, this would all be handled by the software I was using totrack the projectiles, a piece of software made by Cabrillo college which wouldnot only allow me to track the position of the projectile for each frame but alsolet me set a scale for the scene so any measuring I needed to do could be doneon the computer against this scale. The software would also provide me withthe parabola I needed from the video in the form of a polynomial which I couldthen plot against my calculated parabola to compare.

2.3 Velocity Measuring

With the cannon I had got hold of I had one of the values I needed sorted, theangle. This just left me with finding the velocity of the projectile, for which Ihad 2 choices. Either I could do it by assessing the footage or I could do it withhardware.Through software I would be able to use coordinate geometry to find the dis-tance the projectile travelled through a time period and from this calculate thevelocity, the downside to this would be my scale, as discussed previously thefootage from the slow motion camera wasn’t of great resolution meaning anysort of accurate measuring was out of the window (mainly due to blur on theprojectile so not knowing its exact location) which would leave this method offinding distance with great uncertainty. Which is why I opted for the alterna-tive, building a piece of hardware carry out the measuring.

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What I came up with was a light gate style addition to the cannon which wouldmount through the rails on the top and bottom as visible in the above picture.The barrel extension would contain 2 infra red LED traps connected to a microcontroller , as the first gate was broken a value of the time since the start of theprogram running would be saved and then the same thing would happen for thesecond gate. This gave the time taken between the two for which I knew thedistance meaning I could calculate the velocity between these 2 points. I couldthen get instant read off on my laptop of the results. Below is the schematic forthe device , the program I wrote to run it with a description of how it worked,and also a copy of how I received the results from the device.

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3 Risk assessment

With the obvious dangers apparent when using projectiles in an experiment Itook the following precautions to ensure my safety and safety of those aroundme.

• I wore eye protection at all time and asked those in close proximity to myexperiment to do so as well

• A barrier was set up around the area I was working in to stop peoplewandering across the flight path unaware

• People around me were alerted when I would be firing the projectile

• A back panel was put up to stop the projectile travelling outside of thefiring area

4 Journal

4.1 -Lesson 1

Lesson 1 was primarily spent getting the equipment working ,finding a suitablelocation for where I would be carrying out the experiment and how I’d be set up.

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It was after seeing the materials available to me in this lesson that I progressedthe aim of my project from investigating solely air resistance based on shapesof projectiles to a more broader focus due to limited numbers of projectilesavailable to me.

4.2 -Lesson 2,3

Lessons 2 and 3 were used for preliminary testing making sure everyone woldwork as expected and trying to reduce error as much as possible. This is spokenabout more in the ’Preliminary trials’ section.

4.3 -Lesson 4

Lesson 4 was spent putting into place the solutions I had came up with in lesson3 and verifying they would all work. This included the new scale and testingout my infrared barrier.

4.4 -Lessons 5-9

Lessons 5-9 were spent gathering the data , doing 5 repeats for each of the 3power levels for varying angles.

5 Analysis

5.1 Visual analysis

My primary method of analysis is through visual comparison, comparing thegraphs from the video and from calculation. As mentioned earlier I am usingthe tracking software from made by Cabrillo college to track the video. Thisgives me a polynomial for the best fit curve of the parabola, it also allows meto find x and y components for any time value, find acceleration at any pointetc whilst also working under scale set up using my markers in the real scene.With over 60 graphs I couldn’t include them all in the report and so the onesbeing shown are just examples for each power and angle ,typical of what theothers showed.By comparing the various angles it is apparent that with the smaller angles theapproximation hugs tighter the actual curve than those at 60 or 50 degrees, Ibelieve this is due to the fact that with less of a vertical velocity componentthe smaller angles are less affected by initial air resistance on fire, when thevelocity is greatest ,due to the fact that the projectile is spending more time atits lowest speed when travelling at a higher angle.. And then air resistance inthe horizontal direction will affect each projectile equally independent of theirangle.When comparing the different power values it is visible that the highest powerlevel has far more accuracy than the lower power values, and so a velocitytravelling at a higher velocity is a better approximation. This could be due tothe fact that with the high velocities it has a higher kinetic energy, giving it abetter chance to overcome the resisting force of air resistance.One thing we would expect to see with the real life experiments which isn’tso apparent in these results is the shape change we would predict with air

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resistance. What we would expect is to see a parabola not travelling as far andwith a lower maximum point. Whilst we are seeing often a shorter distancetravelled it seems usually that in fact the real world projectiles have a highermaximum point than those which have been predicted.

Figure 1: A predicted model of air resistance on a projectile

5.2 Time of flight

Using the formula t = 2u sin θg I derived a value for the time of the flight of the

projectile. I then calculated the % error between this calculated result and thereal result, found by analysing video frame by frame and then multiplying by1

120 where 120 is the frame rate of the camera. To find a value for those wherethe projectile left the frame before falling back to the same height I tracked theprojectile and then using the best fit parabola found the root of the generatedequation.

The first graph in the time analysis section of graphs shows these errors. Whatyou can see is a quite consistent error for each angle and power of roughly 10%this implies that some correction could be applied to the equation which wouldmake it more accurate, the problem is that whilst the graph doesn’t show whichdirection the error is in by evaluating the raw data table it can be seen thatwhilst mostly an overestimate there was also time where the calculated valuewould be smaller than the predicted value and so in these cases a correctioncould not be applied across each experiment.To understand why most of the predicted values were an overestimate we mustinput air resistance into our model, this would act as a resisting force on theprojectile (opposite to its direction of flight) decreasing its vertical acceleration(as force and acceleration are proportional) meaning a lower greatest height is

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achieved, due to the fact the projectile reaches 0 velocity quicker.This lower greatest height would mean less distance travelled overall rather thanthe height predicted by ignoring air resistance, less distance would mean lesstime taken to get back to the height of projection.

5.3 Range of flight

To calculate the horizontal range of flight I used the formula Srange = u2 sin 2θ2g

derived earlier. Again I calculated the % error for each experiment and plottedthem on a graph. This graph again shows a net constant average error howeverit shows much greater spread than with the time values. The average error isalmost double that of the time of flight at 20%. The recorded values were foundusing similar methods as before. By looking at the calculated distance againstactual distance for each power graphs it seems that in this example it is not theprediction that is at fault. The graph shows some anomalous results in the 40degree experiment where when you would expect this to have an , on average, greater range than that of the 30 degree experiment it fell short, but thenthe 50 and 60 degree experiments followed the predicted trend line. Implyingthat my setup was open to systematic error affecting the results. With thesecalculations it can be seen that the calculated graphs again usually overestimatethe distance. This would again be down to air resistance which would result ina shorter parabola.

6 Evaluation

This experiment was riddled with systematic error. The most major of whichwas the barrel extension I made, whilst it did a good job at measuring velocityand agreed in that sense with every other calculation I did to check its validitythe added length it gave to the barrel caused problems of deflection. Especiallyat the lower power levels the projectile would hit the end of the barrel, firing atan angle different to that which the launcher was set at. Whilst sometimes thiscould be noted and a repeat could be done other times it was less noticeableespecially when it happened at higher velocities. This could be in part respon-sible for the extra lift visible on the real life projectiles.Another theory I have for this unaccounted for lift is the projectile itself, theprojectile was a small wooden ball, wood is prone to having an irregular struc-ture, with small knots throughout it, these irregularities could have put spinon the ball, in a vertical axis along its direction of travel, this spin could easilyresult in the phenomenon of the Magnus effect. Where turbulence is left behindthe projectile causing a lift due to pressure difference, similar to the way planewings work. To counteract this it may have in fact been useful to use a projec-tile method such as air projection, where a tighter barrel would be needed as Icould use rifling like in a gun barrel to add spin in a vertical plan perpendicularto the direction of motion which may have stabilised the projectile reducing theeffect of any irregularities in its weight.The fact also that some of my projectiles were travelling off frame before com-pleting their parabola meant that I was having to extrapolate from what I couldsee, increasing uncertainty. This led to real problems in the 60circ test whereI didn’t have enough tracking points to really find any sort of curve of the

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parabola before it left the top of the screen, and so no data could be found forthese flights.One of the subjects I would like to explore if I was to redo the experiment ismodelling the air resistance on the projectile also and seeing how accurately thiscould be modelled using the formula

Fair =1

2ρACu2

where ρ is the density of air, C is the drag coefficient depending on the shapeof the particle and A is the cross sectional area of the projectile.I would also make an effort to find a more regular and repeatable method offiring which also added spin to the projectile.

6.1 Uncertainties

The theoretical uncertainty of my light gate setup was ±0.000004 seconds (forits timing value), due to this being the cycle rate of the micro controller I wasusing however taking into account the spread of the IR LEDs I was using and sothe uncertainty of when each beam would be broken I took the uncertainty inthe velocity value to be ±0.005ms−1. I believe this to be the lowest uncertaintyI could achieve with the equipment to hand.The major uncertainty was in the angle value, to measure the angle I used thebuilt in protractor and plum line on the launcher, whilst this can be quite anaccurate way to measure it is prone to parallax and also with each fire therewould be a slight movement of the barrel, taking this into account I gave theangle an uncertainty of ±3◦. This could have been improved by using a moresturdy base for my launcher or by finding some other method of measuring theangle, less likely to shift and without parallax.When finding the range of the projectile I gave the distance an uncertainty of±0.02m , this was due to the fact that my scale was made from 0.04m strips,With my timing I managed to reduce the error by using a slow motion camera,shooting at 120FPS rather than the usual 30 or 25 this managed to give me anuncertainty of ± 1

120 th of a second as I was able to go through frame by frameto find when it crossed the axis I had set up at firing height in the trackingsoftware.

7 Conclusion

To conclude my report the experiment has shown that the the approximation ofclassical mechanics whilst not perfect does give a reasonably good model for thereal life motion of projectiles. Especially for smaller angles of firing and highinitial velocities. However I do believe this conclusion lacks in some strengthdue to the sporadic nature of the data I received and often high uncertainties.Even with this taken into account though I believe that for most circumstancesthey provide a suitable model ,unless very high precision is needed.

8 References

• Mechanics 2 for OCR textbook ISBN-10:0521549019

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• http://www.britannica.com/EBchecked/topic/357684/Magnus-effect

• http://www.wired.com/wiredscience/2012/01/projectile-motion-primer-for-first-robotics/

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