Physics Extended Essay

What is the optimum amount of water required to make an angled water rocket fly the furthest?
Name-Harjot Singh Gill
Age – 16
Candidate Number-#3260-0053
Supervisor Name- Mr George
Subject-Physics
School-King George V School
Word Count-3651
Abstract word count-237
Submission Date- 10th June 2013

Table of Contents

Page Number | | 1 | Title Page | 2 | Contents Table | 3 | Abstract and Introduction | 4 | Planning | 5 | Equipment description and setup | 6 | Procedure of experiment | 7 | Variables | 8 | Table showing methods of reading and uncertainties of measure and Analysis | 9 | Analysis | 10 | Analysis | 11 | Analysis | 12 | Table showing uncertainties and errors | 13 | Evaluation and conclusion | 14 | Appendix | 15 | Photo of Apparatus | 16 | Parabolic motion drawing | 17 | Bibliography |

Abstract
For my extended essay, I decided to conduct an experiment; to find the answer to the following question, what was the optimum amount of water required to make a water rocket travel the furthest? What I did in the experiment was adding an initial amount of water (100 ml) to the rocket and measuring the distance the rocket travelled. After taking down the results I repeated the experiment in an effort to find out which value of water was the optimum amount, required to make it fly the furthest, I came to the conclusion that it was 120 millilitres, which made the rocket fly 25.97 meters. My initial hypothesis of the experiment was that the more water that I added to the rocket, the further the rocket would fly. I had set controlled variables as well as differentiated between my independent and dependent variables. The main purpose of me carrying out this experiment was that I wanted to take it along with me to my challenge week trip to Cambodia, to give to the children in the orphanage, so I needed to ensure myself that I knew the maximum distance the rocket travels and the optimum volume of water required to make it travel that distance. My controlled variables was my angle of inclination which I kept at 45 degrees as well as the initial pressure which I added to the water rocket, which was 3 Pascal’s. Word count- 237
Introduction

The main aim of the experiment that I have carried out was to find the optimum amount of water to go into an angled water rocket to make it fly the furthest in a horizontal direction.

The reason for conducting this experiment was that I intended to take it to the Cambodian orphanage that I am going to for Challenge Week. I felt as if it was necessary to know the maximum distance that the water rocket could travel, as when I take it to Cambodia, I wanted to organize a competition amongst the children, therefore it was necessary for me to know the maximum distance that the rocket could go.

The key aspects of the experiment that I kept the same were that I kept the apparatus of the experiment at a fixed position, the angle of inclination (The angle of inclination is the angle between the horizontal ground and the orientation of the rocket) was kept constant and the amount of pressure that I added to the rocket was kept the same.

Planning

This rocket obeys Newton’s Law of Motion; these include Newton’s third law. According to Newton’s Third Law of Motion, the force exerted by the air inside the rocket causes the rocket to exert an equal but opposite force onto the air. Pushing the rocket forward. The critical factors that were taking into consideration for the experiment, were the positions of the equipment, outside temperature, the horizontal velocity of the rocket was constant, pressure applied to the rocket was 3 Pa as well as wind speed was constant. I knew that the wind speed was constant as I used an anemometer, to find out the wind speed and I found that it was more or less constant over a certain range.

The equation of Newton’s law that we used to measure the range, was r= vCos θ x t Where v = velocity, θ = angle of projection, t = time taken for the total flight and r=range, which represents the horizontal distance travelled by the rocket. Since the rocket is projected at an angle its velocity has two components, which is why the rocket follows a parabolic motion (Figure 2, in the appendix), the vertical component is due to gravity, so it is constant, and however the horizontal keeps changing due to air resistance.

My hypothesis was that as you increase the amount of water, the distance travelled by the angled water rocket would gradually increase, because the greater mass inside of the rocket, the greater the specific impulse. Specific impulse is simply the force multiplied by time. Specific impulse shows the instantaneous force applied to the rocket at any moment. By measuring the specific impulse we want to insure that the minimum force applied by the rocket is greater than the force required to keep the rocket in its projectile path.

As we change the volume of water, we observe that the horizontal distance travelled by the rocket kept on increasing, however after a specific volume (120 ml) was added to the rocket, it travelled at its maximum horizontal distance. Due to this if any more volume of water was added, it lead to a decrease in the horizontal distance the rocket could have travelled. As the rocket starts to ascend, the atmospheric pressure outside the rocket is greater than the pressure inside the rocket, creating a large pressure difference, according to Newton’s third law the rocket also exerts an equal but opposite pressure on the atmosphere, which gives a forward thrust to the rocket.

I assumed throughout the course of the experiment that g was constant. Throughout the experiment I also assumed that the position of the set up was unchanged.

Equipment description and setup To carry out the experiment I used a wide range of apparatus as listed below. Pressure gauge- I used the pressure gauge in order to provide pressure force to the rocket. The pressure gauge ranged from 0 Pascal’s to 20 Pascal’s. (Figure 4 in the appendix) Water rocket- The rocket was made of plastic and had polystyrene flaps, it was put at an angle as the hypothesis was to see how far it travels in a vertical direction. (Figure 5 in the appendix) Launch Pad- I used the launch pad to lock the rocket in particular position. (Figure 6 in the appendix) Measuring Cylinder- I used the measuring cylinder in order to calculate the volume of water added to the rocket. Measuring tape- I used the measuring tape to calculate the distance between the initial positions of the rocket to where it finally landed. Water- I used tap water. It had a temperature of 24 degrees Celsius. The viscosity of water is 0.95 mPa/s. The thermal expansion coefficient of water at 24 degrees Celsius is 1.0037. Procedure of experiment 1. I initially set up the launch pad and the pressure gauge in an initial position. I also marked this position with a marker to precisely locate the initial position of the rocket. 2. I then added 10 cm3 of water to the angled water rocket and locked it into place with the help of the pressure gauge. 3. Once the rocket was in place, I pumped 3 Pascal’s of pressure onto the rocket. 4. After applying the pressure to the rocket, I pushed down on the lever attached to the pressure gauge releasing the rocket into the air 5. I then calculated the distance between the pressure gauge and where the rocket had landed after dissension. 6. I then repeated all of the steps listed earlier with different volumes of water to give me a wide range of results in order to make my experiment more reliable. 7. I also did three repeats of the experiment in an effort to increase the precision and accuracy of the experiment.

Variables

Independent Variables

The independent variable in the experiment was the amount of water added to the water rocket. The reason behind changing the volume of water was to see what effect the amount of water had on the ability to make the rocket fly in a vertical direction.

Dependent Variable

The dependent variable in this experiment was the distance travelled by the rocket after it had been fired from the pressure gauge.

Controlled Variables

The amount of pressure added to the rocket- the initial pressure applied to the rocket for every experiment was kept constant; this was to ensure that the results obtained from the experiments were valid. The initial pressure acts upon the center of pressure. Center of pressure is defined as the average location of pressure regarding a system. It can be calculated using the formula,
Center of pressure = (∫x * p(x)dx) / (∫p(x)dx)

Angle of inclination- the angle at which the water rocket was fired was kept at a constant, because theoretically we understand that at a 45o degree angle, the distance travelled by the rocket should be maximum. The way in which I kept the incline of the rocket at a certain angle was locking it into place with the use of a latch system installed in the pressure gauge.

Table showing methods of reading and uncertainty of measures.

Reading | Method of acquiring | Uncertainty | Volume of water | Measuring cylinder | ± 0.5 ml | Pressure | Pressure pump | ± 0.1 Pascal’s | Distance travelled by rocket | Measuring tape | ± 0.1 m | Angle of inclination of rocket | Protractor | ±0.1 o degrees | Analysis

In this experiment a model rocket is being subjected to four forces, weight, drag, lift force and thrust force. At first we fill the water rocket, with a certain amount of water, measured by a standard measuring cylinder, we then lock the rocket into the holder and then apply force to the piston, the results of my experiment are shown in figure 0 in the appendix. The pressure is 3 Pascal’s throughout the experiment. Pressure force and the weight force are responsible for the rocket going up in a vertical direction, since the same angled water rocket is used, it’s only the volume of the water which affects the weight force. Due to this in the actual rocket, the weight force decreases comparatively greater than that of the angled water rocket, hence the distance travelled by that of the actual rocket would be greater than the angled water rocket. The basic trajectory of the angled water rocket was projectile motion, which is a parabolic path followed by the rocket due to gravity (Figure 2 in the appendix). Another way of maximizing the distance travelled, for the optimum volume of water could be setting up the experiment on top of a mountain, so that we reduce gravitational acceleration (g), although the change in (g) value is almost negligible.. We know from Newton’s law of motion that force= mass x acceleration, in this case acceleration is g. With an increase in altitude, g is likely to decrease, so the vertical component of force (Fy), when the rocket is descending will also decrease. So θ will decrease because we know that tan θ = Fy/Fx.. To calculate the range travelled by the rocket on top of the mountain, the equation used is, R = vCos θ x t. Throughout the duration of the experiment, a set value of 3 Pascal’s of pressure was used, due to the fact that I was carrying out this experiment within a limited area. However if I was allowed to carry out the experiment in a greater area, I could have increased the amount of pressure which subjected on the rocket, then I could have observed how the distance travelled by the rocket was affected by the change in pressure. From a physics hypothesis, we know that p=F/A, where pressure =p, force =F and A = area, so if pressure increases, force increases, again going back to Newton’s second law, where F= ma, where m= mass and a = acceleration, we know that with an increase in force, the acceleration will increase, therefore the object will travel further. The amount of pressure will not be a problem for the children in Cambodia, because as they would not have limited space to use the rocket, therefore allowing them to add more pressure.

In order for the rocket to follow its supposed projectile motion, it is necessary that it be kept stable throughout its flight. It is necessary to keep the rocket as stable as possible so that it can ascend to a greater height. The rocket needs to fly at a certain height or else it will hit obstacles in its path, such as houses with trees, which will prevent the smooth projectile path of the rocket. There are certain principles that must be maintained, the most important of all is that the center of gravity should be infront of the center of pressure (where the pressure is applied) . In an normal water rocket, the center of gravity is usually below the center of pressure, causing the water rocket to be unstable, in this case we use air flaps to bring down the center of pressure, below the center of gravity, but this turns out not to be such an effective method, so we try to modify the design of the rocket, where we have a thin cylinder with a smaller diameter at the lower end and a wider diameter toward the nose of the rocket, due to this the amount of water is greater towards the top of the rocket, as compared to the bottom, meaning that there is a greater weight added towards the nose, thus the center of gravity shifts upwards, making the rocket highly stable.

Even though the speed of the rocket seems to be constant at first, if you notice carefully, it is actually changing over time. In fact it is decreasing due to the air resistance, in that case using a formula, distance = speed x time, will give an estimated value of distance, rather than the actually value of distance. In order for the distance to be more accurate, we must find an alternative way of measuring the distance. Such a method could be by dividing the span of time taken for flight into short time intervals. Now each of these time intervals corresponds to a speed, so if you multiply the respective speed with the respective time intervals then we can get an approximated distance in each interval. Finally we can sum up the distances at each of these intervals to get the total distance travelled by the rocket. The main principle of this method is that the time interval is very short so the speed of the rocket stays more or less constant. This method is slightly more accurate than rather than assuming that there is a constant velocity throughout the duration of the experiment.

However, the most up to date method of measuring the distance travelled of a rocket whose speed varies with time would be integration. Let us assume that f(x) (function of x) represents the speed of the rocket as it varies over time. A graph showing the speed varying with time will be shown in the Appendix, the area under the curve would represent the distance travelled by rocket, so all we need to do is to find the area, giving us an accurate measure of distance. A clever way of finding the area could be dividing of the distance a and b into several segments, then let each of the segments be dx. Each of these segments corresponds to a certain value of f(x), which we could name as y. In order to find the area under the curve, we must find the area of each segment and the summation of the area of each segment equals to the total distance travelled by the rocket. The area of each segment can be considered that of a rectangle, we know the formula of the area of a rectangle = length x breadth, which in this case = y x dx. After calculating the area of all the segments, we use the integration equation which is ∫ ydx from a given range of a to b to find the total area, which represents the total distance travelled by that of the rocket. The way that this method links back to the experiment is that it allows us to calculate theoretically the distance travelled by the rocket, we can then use the theoretically calculated distance in an effort to compare the results obtained from the experiment. The reason behind the comparison is that if the results between the theoretical value and the experimental value are similar, this further outlines the validity of the results, but there is also a scenario where the results are different, then we may conclude that there are other factors that have affected the experiment.

The following theories used in aeronautical engineering. There is also a scenario, where the children at the Cambodian orphanage may choose to change the size of the rocket, in order for this to happen, we must maintain certain similarities – kinematic, dynamic and geometric. For geometric similarity there should be similarity in shape, as well as all corresponding lengths of the model and the prototype are in one ratio, this ratio is called λ L, this ratio is equal to LP divided by LM, LP is the corresponding length of the prototype and LM is the corresponding length of the model. Their corresponding angles must also be the same. We also need to maintain kinematic similarities, kinematic similarities is the relationship between two fluid flow systems in which corresponding fluid velocities and velocity gradients are in the same ratios at corresponding locations, for this all corresponding speeds and accelerations are in one ratio. Lastly we need to ensure dynamic similarities; dynamic similarities are a relationship between two separate mechanical objects such that proportional changes in length, mass, and width and breadth in one system go identically. Which means all the forces experienced by the model and the prototype should be in one ratio. For example if we choose a ratio of 1 is to 5 then if the weight force on the model is 15 Newton’s then the weight force of the prototype is 250 Newton’s, same is the case as the three other forces that are subjected upon the rocket.

Table showing uncertainties and errors

Reading | Method of acquiring | Uncertainty | Volume of water | Measuring cylinder | ± 0.5 ml, it was the accuracy of the measuring cylinder that had been used | Pressure | Pressure pump | ± 0.1 Pascal’s | Distance travelled by rocket | Measuring tape | ± 0.1 m | Angle of inclination of rocket | Protractor | ±0.1 o degrees |

Evaluation and Conclusion

While carrying out the experiment, the readings were subjected to a lot of errors. There are two types of errors, systematic errors and random errors, systematic errors due to a mishap in the method of observation, a miscalibration of the instruments used during the experiment or an incorrect set up of the experiment. Taking several readings and then averaging out cannot overcome systematic errors. Random errors occur due to a physical limitation in the person or the instrument observing or undergoing the experiment. There are substantial errors that occurred in this experiment, which tampered with the validity of the results, had these errors not been present it would improve the accuracy of the results, as well as shedding more light on a rather unresolved issue. The sources of error in this experiment are,

1. Reading the incorrect meniscus of the fluid- Usually water has the property of wetting the surface of the container, thus having a concave meniscus. While the observer is taking the measurements, it is important that he reads the lower meniscus to measure the volume. Otherwise, the observer will get a wrong reading of the volume. 2. The position of the apparatus- Every time I fired the rocket, the apparatus moved backwards, but the distance was measured from the set mark that I had laid out. Due to this there is a discrepancy in the true value of the distance travelled with the measured value. This specifically is an example of a random error. The way to improve this error is to measure from the new position of the setup to the point where the rocket landed because it would give more accurate results. 3. Parallax error- while measuring the volume of water, using the measuring cylinder it was important to keep the observers eye in line with the scale of the measuring cylinder. Otherwise there would be parallax error, which would affect the validity of the measurement. 4. Distance measured to the point where the rocket landed- the accuracy of the measuring cylinder was ± 0.1, but in some cases the rocket would land in the intervals between measurements, hence causing me to estimate the distance it had travelled. In order to improve upon the error, I could add a sensor to the rocket in order to more accurately measure the distance travelled. I could also use a laser range finder, which would detect the sensor and calculate the distance from the starting position to where the rocket finally landed. 5. Wind affecting the rocket- if the rocket travels with the direction of the wind, the rocket will move further because the net force on the rocket will be greater however this will give inaccurate results

Appendix

Amount of Water (ml) | Distance travelled (m) | 100 ml | 20.42 | 110 ml | 23.12 | 120 ml | 26.27 | 130 ml | 24.68 | 140 ml | 22.39 | 150 ml | 21.47 |

Amount of Water (ml) | Distance travelled (m) | 100 ml | 20.37 | 110 ml | 23.65 | 120 ml | 25.97 | 130 ml | 24.32 | 140 ml | 22.12 | 150 ml | 20.84 |

Amount of water (ml) | Average distance travelled (m) | 100 ml | 20.40 | 110 ml | 23.39 | 120 ml | 26.12 | 130 ml | 24.50 | 140 ml | 22.26 | 150 ml | 21.16 | (Figure 0 ) (Figure 1)

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Bibliography

http://www.aerospaceweb.org/question/aerodynamics/q0151.shtml

http://www.grc.nasa.gov/WWW/k-12/rocket/rktbflght.html

https://en.wikipedia.org/wiki/Integral

http://library.thinkquest.org/07aug/00983/rockets.html