Physics A2 Practical Coursework Investigating the effect of changing the volume of water on the force, acceleration and velocity of a water rocket Water rockets work by filling an airtight bottle with water and increasing the pressure inside the bottle by pumping air into it until the bottle is fired into the air. The bottle will reach a fixed pressure for every volume of water as the valve for the bottle will disengage at the same pressure every time. This will provide a thrust downwards which will accelerate the rocket off the floor. There will be a force downwards generated by the momentum of the ejected fuel which will increase the momentum of the rocket upwards. However, for greater volumes of water there would have to be a greater force as it would also be accelerating a greater quantity of fuel as well as the bottle. My coursework investigates whether the initial volume of water has any effect upon the force, peak acceleration or velocity of the bottle. Hypothesis: The smaller the volume of water, the greater the peak velocity of the rocket and therefore the greater the peak acceleration and force of the rocket. Initial considerations from preliminary testing Bottle volume: I had three possible bottles to use for my experiment 300ml, 2 litres and 3 litres. 100ml of water was the most effective in the preliminary testing because of the precision of the jet and the speed of discharge, as I was aiming for all of the water to be ejected from the bottle before it left the camera frame. I therefore decided to use the 300ml bottle. Volume intervals: I initially decided to use 15ml intervals of water from 15ml to 300ml as this would give me 20 results which would provide a suitable amount of data to analyse. However, when I started to experiment I discovered that 15ml intervals made negligible difference and therefore decided instead to work with 30ml intervals which would give me 10 results without repeats. This was further confirmed by my discovery that the minimum effective volume was 30ml. Set up and method Equipment: 300ml bottle, 1000fps camera on a ten second timer, camera stand, bicycle pump, rocket wings and nozzle attachment, 300ml reserve bottle. I set up the bottle and pump as shown in Figure 1, ensuring that the bottle could be seen at the bottom of the camera frame from its position on the stand. I also had to ensure that the rocket was placed vertically, although I later discovered that (perhaps due to the position of the pump) the bottle had a tendency to take off at an angle and therefore set the bottle to compensate. I also repeated each volume of water three times in order to make my results more reliable. Risk Assessment My main concern whilst conducting this experiment was ensuring that there was no one in the immediate vicinity to the falling rocket, as when it returned from its peak altitude it was travelling quite quickly and the rocket had sharp edges. I therefore cordoned off the area in which I was working and ensured there was a large possible safe space in which the rocket could land. As I was using the school field, I also checked with the Sports teachers that the area I was using was not close
to any running PE lessons at the time. As the field is close to a road, I ensured that there were no cars on the road at the point at which I set off the rocket so as not to surprise any drivers. I attempted to angle the take-off of the rocket away from any buildings and paths where it could cause damage. I also wore a hard-helmet to avoid injury to myself. Bicycle pump Water bottle and rocket wings Location of camera and stand Method Figure 1: Set up of experiment I filled the bottle with the required volume as measured out from a larger bottle by a measuring cylinder. I then attached the wings and nozzle that connected the bottle to the pump and place the bottle in the same place each time. I discovered that for lower volumes the camera should be placed at a rough distance of 5-6m away in order to ensure the whole jet of water was captured within the frame. I would then check the camera s position, start the ten second timer and at the end of the ten seconds begin to pump until the rocket took off. Between tests I dried the nozzle that attached to the bottle in order to ensure a fair test. I was initially intending to mark the starting level of the water onto the bottle before I set it off in order to use it as a marker on the video. However, upon analysis of the videos I discovered that the water level of the bottle is relatively easy to see without the marker, which was not very clear, and therefore I used the juncture of the bottle neck as a marker point. Similarly, I decided not to use coloured water as the water jet and the bottle was easy enough to see without it. When I began to video the greater volumes of water, I discovered firstly that the water jet was too high to be able to fit into the frame with sufficient detail. I therefore decided to take the videos at 240 frames per second as opposed to 1000 as this still gave sufficient detail whilst allowing me to fit as much of the jet into the frame as possible. I also discovered that volumes of greater than 240ml were unreliable when taking off, and therefore restricted my testing window to 30ml to 240ml. I also had difficulty in ensuring that the rocket took off at precisely 90 o to the ground every time, so instead allowed any data in which the rocket took off at between 45 o and 90 o and the water jet was all expelled before the rocket left the frame.
Data Using Tracker I used the software Tracker to analyse my videos in order to generate the data on velocity and acceleration. I entered the number of frames per second and the number of frames I wanted the video to skip in most cases 10 but for the higher volumes 5 because of the changed number of frames per second. I then created a point mass or point of reference and tracked its progress throughout the take-off of the rocket until it left the frame. Tracker then Figure 2: Screenshot of Tracker program automatically generated the acceleration and velocity data for the rocket which I then exported to an Excel spreadsheet as shown in Figure 3. Data Analysing the data To begin with, I amalgamated all of the repeat data onto one spreadsheet. I then created graphs for each volume of water showing the three repeats see Figure 4. Some of these graphs were more similar than others, perhaps due to differing conditions for some of the repeats as I took them on separate days. In general, as the volume of water increased the data repeats became less and less similar, perhaps due to the effect of a lower frame rate or the fact that not all of the water had been expelled from the bottle by the time it left the frame. Because the video could not be rotated, tracker exported the data as being negative. However, this makes no difference to the data itself and was corrected for further analysis. t x v_{x} a_{x} 0.00E+00-8.41E-02 3.34E-01-8.41E-02-4.50E-03 6.67E-01-8.71E-02-4.50E-03 3.86E-03 1.00E+00-8.71E-02 0.00E+00 7.71E-03 1.33E+00-8.71E-02 0.00E+00-8.10E-02 1.67E+00-8.71E-02-4.73E-02-3.53E-01 2.00E+00-1.19E-01-2.30E-01-6.40E-01 2.34E+00-2.40E-01-4.89E-01-6.57E-01 2.67E+00-4.45E-01-6.75E-01 3.00E+00-6.91E-01 3.67E+00-1.17E+00 Figure 3: Table to show imported data from Tracker. Having taken results for each volume of water, I then took an average for each volume and plotted these averages on a graph see Figures 5 and 6. The yellow highlighting signifies the six results leading up to the peak acceleration. Because the videos lasted for differing lengths of time, each data series was slightly out of sync with the next.
Acceleration (ms-2) 0.00E+00 Time (s) -1.00E+02-2.00E+02-3.00E+02-4.00E+02-5.00E+02-6.00E+02-7.00E+02 A graph to show the acceleration of a -8.00E+02 60ml bottle Figure 4: Graph to show the acceleration of a 60ml bottle over time -9.00E+02 30ml Repeat 1 Repeat 2 Repeat 3 Average Corrected Average 1.89E+00 1.89E+00 1.89E+00-1.89E+00-1.89E+00 1.89E+00 0.00E+00-9.44E-01-4.72E-01 4.72E-01 0.00E+00 9.44E-01 4.72E-01 4.72E-01 0.00E+00 1.89E+00 4.29E+00 2.06E+00 2.06E+00 0.00E+00 0.00E+00 8.59E+00 2.86E+00 2.86E+00 0.00E+00 0.00E+00-9.01E+01-3.00E+01 3.00E+01-2.36E+02-1.62E+02-3.93E+02-2.64E+02 2.64E+02-6.13E+02-4.23E+02-7.13E+02-5.83E+02 5.83E+02-7.50E+02-4.31E+02-7.32E+02-6.38E+02 6.38E+02-5.27E+02-1.81E+02-3.54E+02 3.54E+02-2.67E+02-2.67E+02 2.67E+02-1.18E+01-1.18E+01 1.18E+01 Figure 5: Table to show repeated data with averages from Tracker
Acceleration (ms -2 ) Acceleration (ms -2 ) I then aligned all of the data series so that they peaked at the same point so that the peak acceleration for each volume could be compared. From this graph it is possible to see that the lower the volume of water, the higher the peak acceleration seems to be as shown in Figure 7. 700 Investigating the effect of changing volume on average acceleration over time 600 500 400 300 200 100 30ml 60ml 90ml 120ml 150ml 180ml 210ml 240ml 0 Figure 5: Graph to show the average acceleration of each volume over time. 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Time period (s) Figure 6: Graph to show the average acceleration of each volume over time. 700 Investigating the effect of changing volume on average acceleration over time when results are aligned 600 500 400 300 200 100 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Figure 7: Graph to show the aligned Time average period acceleration (s) of each volume over time. 30ml 60ml 90ml 120ml 150ml 180ml 210ml 240ml
Peak acceleration (ms -2 ) I then plotted the peak acceleration for each volume onto a graph to find if there was any correlation between volume and peak acceleration. Figure 8 shows an inverse quadratic relationship between peak acceleration and mass (or volume.) Although there is a linear relationship between mass and acceleration (acceleration = force mv ), the force depends on the momentum. (F = ) As the mass t bottle accelerates the velocity increases, but the mass of the bottle decreases as water is expelled. It was not possible for me to assess the water level in the bottle in every frame, and therefore I could not examine the video to see if there was a relationship between acceleration and mass in order to work out the force. The trendline fits better as a quadratic line, although a linear trendline fits only slightly worse. I have also extrapolated the trendline until it meets the x and y axes. The point at which the trendline meets the y axis would suggest the peak acceleration when there is no water in the bottle would be just over 700ms -1, and the point at which the trendline meets the x axis would suggest the volume of water at which the bottle would no longer take off would be around 270ml. Unfortunately I was not able to continue the experiment enough to see whether either of these figures are reliable, but it would provide an interesting future experiment to attempt to advance the volumes of water used in either direction to find out. 800 Figure 8: Graph to show the peak acceleration against volume. 700 600 500 400 300 Investigating the effect of volume of water on peak acceleration 200 100 0 y = -0.0042x 2-1.358x + 705.1 R² = 0.9507 0 50 100 150 200 250 300 Volume of water (ml)
Investigating peak velocity using the Trapezium rule 30ml Figure 9: Tables to show the working and data for the trapezium rule for 30ml. Excel dues not X values Y values Trapezium rule X values Y values Trapezium rule allow a set number of decimal places on the formulas view to be specified. 0 0.00 0 0 Change in velocity 0.01 according 1.89 to tracker... 0.01 0.01 1.89 =0.5*(C4+C5)*(B5-B4) 0.02 1.89 0.02 0.02 1.89 =0.5*(C5+C6)*(B6-B5) 30ml (1) 25.45 0.03 0.47 0.01 0.03 0.472 =0.5*(C6+C7)*(B7-B6) 30ml (2) 25.00 0.04 0.47 0.00 0.04 0.472 =0.5*(C7+C8)*(B8-B7) 30ml (3) 22.50 0.05 2.06 0.01 0.05 2.06 =0.5*(C8+C9)*(B9-B8) Average: 24.32 0.06 2.86 0.02 0.06 2.86333333=0.5*(C9+C10)*(B10-B9) 0.07 30.03 0.16 0.07 30.0333333=0.5*(C10+C11)*(B11-B10) 0.08 263.67 1.47 0.08 263.666666=0.5*(C11+C12)*(B12-B11) 0.09 583.00 4.23 0.09 583 =0.5*(C12+C13)*(B13-B12) 0.1 637.67 6.10 0.1 637.666666=0.5*(C13+C14)*(B14-B13) 0.11 354.00 4.96 0.11 354 =0.5*(C14+C15)*(B15-B14) 0.12 267.00 3.11 0.12 267 =0.5*(C15+C16)*(B16-B15) 0.13 11.80 1.39 0.13 11.8 =0.5*(C16+C17)*(B17-B16) 21.51 Total =SUM(D5:D17) 30ml Total Figure 10: Table to show the velocity according to tracker. The trapezium rule works out the area under a graph. As the area under an acceleration-time graph represents the velocity, I decided to compare the peak velocity suggested by tracker to the peak velocity from the trapezium rule see Figure 12. Volume (ml) Peak Velocity Tracker Trapezium rule 1 2 3 Mean 30 21.51 25.45 25.00 22.50 24.32 60 *30.24 28.70 27.90 29.80 28.80 90 *38.99 27.50 31.60 25.70 28.27 120 *29.03 27.80 30.20 25.10 27.70 150 21.88 25.10 26.40 22.40 24.63 180 *13.31 23.50 19.80 23.60 22.30 210 *40.34 25.80 19.20 30.00 25.00 240 *10.45 10.90 14.10 21.40 15.47 Figure 11: Table to show the velocity according to tracker
Acceleration (ms-2) Peak velocity according to Tracker (ms-1) Upon initial inspection of the table, there seems to be very little effect on the peak velocity by changing the mean. It was difficult to come up with a reliable value for the trapezium rule as often the bottle had left the frame before its acceleration returned to 0, and so a certain amount of extrapolation was involved in order to work out the velocity for a full curve. This is denoted by a * next to the values which have been extrapolated see Figure 13. 35.00 30.00 25.00 20.00 15.00 10.00 5.00 0.00 0 10 20 30 40 50 Peak velocity according to the Trapezium rule (ms-1) 800 600 400 Figure 12 A graph to compare peak velocity as derived by Tracker, and peak velocity as derived by the Trapezium rule. Trapezium rule graph for 90ml Extrapolated acceleration Original acceleration 200 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 Time (s) Figure 13 A graph to compare extrapolated acceleration against original data Figure 12 seems to show very little correlation between Tracker peak velocity and Trapezium peak velocity, especially towards the later data points. As shown by the trendline on the graph, there should be a 1:1 correlation between the Tracker data and the data from the Trapezium rule as both come from analysis of the same videos. Of the two, the Tracker data is likely to be the most reliable as it is generated with very little room for error, whereas the Trapezium rule data involves a large
Peak velocity (ms-1) amount of extrapolation due to the absence of some data from the acceleration graphs. This is later shown by the graphs of peak velocity for both methods against volume see Figure 14. 45.00 40.00 35.00 30.00 25.00 20.00 15.00 10.00 5.00 0.00 0 50 100 150 200 250 300 Volume (ml) Tracker Trapezium Figure 14 A graph to compare peak velocity as derived by the Trapezium rule and by Tracker against volume of water in the bottle. There seems to be very little obvious correlation between volume and velocity for both data series. However, Figure 14 does show one seemingly anomalous data point at 210ml. When this data point is removed, Figures 15 shows a much closer correlation to the suggestion that the peak velocity is greatest at between 50ml and 100ml. The force acting on the bottle depends on the change in momentum over time (F = mv ). As the velocity increases to a peak at a relatively low mass, it t would be safe to say that the momentum of the bottle is likely to be greatest at lower volumes of water. However, as I have no means of ascertaining the volume of water left in the bottle at peak velocity, I am unable to work out the momentum of the bottle in order to see whether the force changes on it. The velocity may be low for larger volumes of water because it would take a greater force to accelerate the rocket away from the floor as the mass is greater, and the rocket would therefore travel more slowly as a result. However, this would suggest that the lowest volume of water should represent the highest velocity, which is not the case. Could it be that there was not sufficient propellant in the lower volumes of water to allow the bottle to reach a peak velocity?
Peak velocity (ms-1) 45 40 35 30 25 20 15 10 5 R² = 0.9463 Trapezium Tracker Poly. (Tracker) 0 0 50 100 150 200 250 300 Volume (ml) Uncertainties Figure 15 A graph to compare peak velocity as derived by the Trapezium rule and by Tracker against volume of water in the bottle when anomalous data is removed.
Peak acceleration (ms -2 ) 700 600 Investigating the effect of volume of water on peak acceleration 500 400 300 200 100 0 0 50 100 150 200 250 300 Volume of water (ml) Figure 16: Figure 8 with uncertainties The main source of uncertainties in my experiment was my use of the program Tracker. Because I measured the volume of water for each test using a measuring cylinder with a scale of 10ml, my maximum uncertainty for volume is 1-2ml, and therefore negligible on a graph. I derived the uncertainty for Tracker by using the measuring scale I had already implemented by measuring the water bottle to work out the size of one pixel in real life. I assumed there would be a one pixel uncertainty either side of the tracker icon I was placing. This worked out at a 3.2cm uncertainty for distance, and therefore a 6.4cm uncertainty for velocity and a 12.8cm uncertainty for acceleration. As shown in Figure 16, some of the error bars on the data points do overlap and this would suggest the data is not wholly reliable. However, the small size of the error bars coupled with the negligible overlap and the general trend of the data would suggest that there is a definite inverse relationship between volume of water and peak acceleration.
Peak velocity according to Tracker 50.00 45.00 40.00 35.00 30.00 25.00 20.00 15.00 10.00 5.00 0.00 0 50 100 150 200 250 300 Volume (ml) Figure 17: Figure 15 with uncertainties Whilst an uncertainty of 12.8cm had a relatively minor effect on the reliability of Figure 8, an uncertainty of 6.4cm had a much greater effect on the reliability of Figure 15. As can be seen in Figure 17, all of the standard error bars overlap almost completely for both data sets and this suggests that they are quite unreliable. Their unreliability could be reduced by taking further repeats and ensuring that the uncertainty whilst using Tracker is reduced. Conclusion The aim of this experiment was to address several questions. They are as follows; - Does the volume of water in the bottle affect peak acceleration? - Does the volume of water in the bottle affect the shape of the acceleration curve? - Does the volume of water in the bottle affect the peak velocity? - Are the results self-consistent? Does the volume of water in the bottle affect peak acceleration? As shown in Figure 8, the higher the volume of water, the lower the peak acceleration. This data seems to be very closely correlated around an exponential trendline, which suggests that there are other factors affecting the acceleration of the bottle apart from force and mass. Because we do not know the mass of the bottle at 0.001 or 0.04 second intervals, I am unable to ascertain whether the mass at each point has a bearing on the velocity or acceleration. Does the volume of water in the bottle affect the shape of the acceleration curve? For the lower volumes of water, the acceleration of the bottle over time is largely symmetrical and rises steeply from a near-zero value to peak. This would suggest that the bottle accelerates very suddenly and very quickly. For the greater volumes of water, the acceleration of the bottle increases far more gradually and is less symmetrical, suggesting that it takes longer for the bottle to reach a peak acceleration, as seen in Figure 7.
Does the volume of water affect the peak velocity? As shown in Figures 15 and 16, the peak velocity seems to increase as the volume of water increases between 30 and 90ml to a maximum, and then decreases once more. Because of the lack of available other recorded variables, I am unable to ascertain why this is although it could be suggested that it is due to needing a minimum mass of propellant to reach a peak velocity. Are the results self-consistent? The results that compare peak velocity due to tracker to peak velocity from the trapezium rule would suggest that there is a difference between the two generated values. I would suggest this is because the peak velocity generated by the trapezium rule is less reliable due to a large amount of extrapolation being involved. Because I repeated the experiment three times for each volume of water, most other results seem to be reasonably consistent in values. There was a slight change in experimental data on some of the second and most of the third repeats due to my using a different frame rate on the camera when recording the experiment. Suggested further experiments - Looking more closely at the volume of water during each timeframe to investigate whether there is a change of force for each water bottle. This would be difficult as moving the camera back far enough to ensure that all of the water has been ejected from the bottle by the time the bottle has left the frame would mean that there would not be enough detail on the video to see the water level. Similarly, however, bringing the camera close enough to see the water level in the bottle would mean that not all of the water would have been ejected from the bottle by the time it left the frame, and this would have a possible negative effect on the results. - Looking at whether changing the density of water (such as adding salt) would make a difference to the acceleration of the bottle or the force acting on it. I tested this during my experimental data and saw no visible effect, but this could potentially draw a link between force and density. Similarly, an experiment could also be tried by using lemonade as fuel, as lemonade contains air bubbles and therefore could affect the pressure in the bottle.