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Lab Report The Skydiver [Skydiver] Daan Rijpkema T4Y - TTO4 Physics (Moderated) October 2008 Word Count: 3641 Table of Contents Introduction......................................................................................................................................... 3 Course of Action .................................................................................................................................. 3 Chapter 1 – Velocity and Gravity ................................................................................................... 3 1.1 Terminal Velocity ..................................................................................................................... 3 1.1.a Hypothesis ........................................................................................................................... 3 1.1.b Variables .......................................................................................................................... 4 1.1.c Plan ...................................................................................................................................... 5 1.2 Conducting the Experiment ..................................................................................................... 5 1.2.a Terminal velocity and Mass ................................................................................................. 5 1.2.b Gravitational Force and Mass .......................................................................................... 7 1.3 Evaluation ................................................................................................................................ 8 Chapter 2 – Parachute Landing ..................................................................................................... 9 2.1 Friction Constants.................................................................................................................... 9 2.1.a Hypothesis ........................................................................................................................... 9 2.1.b Variables ........................................................................................................................ 10 2.1.c Plan .................................................................................................................................... 10 2.2 Conducting the experiment and calculations........................................................................ 10 2.3 Questions............................................................................................................................... 12 Bibliography ............................................................................................................................... 16 Printed Sources ................................................................................................................................. 16 [Page 2] Introduction A skydiver jumps out of a plane. His height is 2000 meters from the ground. The mass of the skydiver is 100 kilograms. He uses the parachute after 30 seconds of falling. To avoid breaking any limbs or damage himself he has to be slowed to a safe velocity. In addition the para- chute should not slow down the person too quickly as this can also cause injury to the skydiver. Gravity Forces Distance In this lab report I will research and simulate two experiments and use these Time to answer and solve various questions using vectors and elements like the Kinetic Energy ones to the right. Velocity Air Friction Course of Action Acceleration Using the general scientific investigation structure in which I will state my problems, make a hypothesis, find and define variables, plan an investigation, execute it and evalu- ate it. I will use the program Vissim to simulate the problems and enhance my calculations. This will also give me in depth graphs that will help me explain my reckoning. Chapter 1 – Velocity and Gravity 1.1 Terminal Velocity In these questions I will be using the details about the skydiver given in the introduction. The subject of this chapter is about terminal velocity and gravitational force. The guiding question can be found below. Does mass of the skydiver affect the terminal velocity? I will start by writing a hypothesis and defining my variables and plan. Afterwards I will execute the plan and present Skydiver - 100kg the results. Accelerating 1.1.a Hypothesis with gravity - 9.81 m/s2 I think that the answer to the question is yes: the mass will affect the terminal velocity. It is quite obvious that a more massive person or object has a faster vertical descent. This Force of gravity 1000 N or 981 N can be tested by having three different weighing skydivers jump out and record their velocity and acceleration, and thus terminal velocity. An alternative can be simulating the above experiment, which is what I am going to do. [Page 3] From calculations of physical mechanics, I also know that the formula of mass can also be rewritten as Ns2/m or Newton times Second Square divided by meter I know that: The formula of Newton’s Second Law: Here I can see that Mass can also be formulated when looking at the force divided by the accelera- tion. When turning Newton’s formula around: This proves that Mass is a variable of acceleration, together with the above fact that kilograms is equal to Newton multiplied by speed squared1 divided by meter I know that acceleration defines the velocity curve of the skydiver (how quickly his falling velocity increases) and that if mass affects acceleration which affects velocity, therefore the terminal velocity [Physics for you] is also affected (see paragraph 1.2.a). The only thing remaining is to experiment with it to create more substantial proof. 1.1.b Variables The variables in this experiment of finding out (terminal) velocity are: What do I change? - Independent The mass of the skydiver - kg This will be changed in order to find potential results. As I am looking for the effect of mass on velocity, this is my independent variable. The Gravitational Force - N This is proportional to the mass and will therefore also change if mass is changed, and thus is a variable. 1 Which is an accelerating speed [Page 4] What do I observe? - Dependent The Terminal Velocity and average velocity - m/s This is what I am trying to find out and will be visible in the graphs (see below for more infor- mation on terminal velocity). The time it takes to reach this also plays a role in observations. What do I keep the same? - Controlled The friction constant - the k in (formula for air friction) This will be kept at one number as I am trying to make a realistic simulation without deviating results. This is also sensible because I am only testing the free fall (no parachute) part of the skydiver's flight. If I would change this, my results would lose accuracy. 1.1.c Plan In this lab simulation I will try to prove whether mass affects terminal velocity: I will start with a short introduction about what terminal velocity actually is. Then a test in which I simulate several masses and their falling velocities. Then an additional test in which I try to perceive if mass is a variable in gravitational force (and thus if it affects terminal velocity as well). 1.2 Conducting the Experiment 1.2.a Terminal velocity and Mass To answer the question I need to start off finding out what terminal velocity is. The mass of the sky- diver, 100kg, exerts a force of 981 N at the start of the jump. His mass makes him fall at a constantly increasing velocity (which is called acceleration)2. This is caused due to gravity pulling the skydiver to the ground. Air Friction opposes movement, thus causing it to increase as an object moves faster. This applies to the skydiver as well. As its velocity increases so does his air friction. This looks like graph 1. The skydiver jumps and rapidly increases in velocity in the first eight or so seconds. After this his forces become balanced and thus neutralizing acceleration and thus a constant velocity. This is terminal velocity: the velocity will not increase and is at its Terminal Velocity highest possible value. The terminal velocity is achieved when the gravitational force is equal to the air fric- tion force and is thus neutralized. In the graph this is happening after around 18 to 20 seconds or so, at a velocity of about 55 m/s. 2 Velocity is the speed and direction of an object. Acceleration is the increase in speed per second. Gravity is 2 pulling on an object with an accelerating velocity of 9.81 m/s , causing all objects to have an acceleration of 9.81 meter per second squared (without friction). [Page 5] But would mass have affected this element? If the skydiver would have had a mass of 50kg or 75kg and thus exert a force of respectively 500N and750N, the outcome would look like those in graph 2. This shows that as mass changes, the acceleration changes and therefore the terminal velocity is affected. This is quite logical: as the mass increases, so does the gravity pulling on the larger mass, thus increasing the acceleration, thus increasing the air resistance opposing the acceleration. In other words, the higher the mass, the higher the acceleration, the longer it takes to reach terminal velocity as the air friction needs more time to oppose the gravitational force and acceleration. The calculation used for the graphs above can be found in figure 1 below, which is a visual represen- tation in the program Vissim. The diagram shows that mass is a variable (visible in the simulation calculation elements) and thus affects the outcome. The data below is used for a skydiver with a 100kg mass. [Page 6] 1.2.b Gravitational Force and Mass Here I will experiment changing the mass and gravitational forces to find out by keeping the k values constant. Or just Calculating gravitational force gives figure 2 above in Vissim. I could add a range of mass and gravita- tional forces to find possible terminal velocity differences, but this is not necessary to show the influ- ence. It is really quite simple: when looking at figure 1 and 2 and section 1.3 I know that mass affects ter- minal velocity. If I then look at the question whether gravitational force then also affects it, I can give the answer: figure 2 shows that gravitational force varies with mass (it has mass in its formula). Therefore if the gravitational force would change, so would the mass. And as a result gravitational force also affects terminal velocity. An example can be found below in graph 3. [Page 7] 1.3 Evaluation At the start of this chapter I thought the whole would be hard. However, using all the information given and looking at it from a simpler perspective than expecting a very hard question yielded great results. In my eyes I’ve been clear in my revelations and got great help from Vissim and the teacher. Due to the use of Vissim my results were accurate and calculations could easily be repeated and/or tested. The only way to make errors with it is by inserting conflicting values, in contrast to real life where this experiment would have been hazardous to perform accurately and extensively. With this I can prove my hypothesis of which I was almost certain that it was correct already. This has increased the strength of the statement and helped me understand what terminal velocity is and how it is affected. Because of this I think the experiment has been very successful and correct. I am confident I did not make major and critical mistakes or I would have seen this in the number of graphs above. Due to the fact I also posted my calculations I will be able to see what I did wrong right away when noticed. Improvements to this experiment could be made by adding more variables to test this specific state- ment. There are probably more variables affecting terminal velocity that were not discussed in this project, yet would’ve been very interesting. Next to that Vissim is capable of much more students do not know yet: this might be of use in opti- mizing simulation and thus the interactive educational value of this project. This might require addi- tional time and learning (might require pretty advanced techniques but would possibly yield better results and outcomes). 3 If I would continue the investigation the first thing I would do is research more ways to enhance the project (see previous paragraph about improvements). Next to that I might try new methods of simu- lating this or repeat it in a similar way to improve realism. Research on specific mass values or k val- ues and patterns in these results (for example the formula of the graphs on page 7) can be added. This is because in reality a skydiver is also affected by variables such as surface, material and area of the object, the air density and movement/position while skydiving. A possibility would be to add a more realistic, worked out k value with more of the above values in it (or another value to affect the results as a separate variable in the plot). This would give me a more reality based result and opti- mizes simulation (and this is probably possible in Vissim). 3 Do note that this is just speculation and opinion: I am not sure of this statements’ actual value. [Page 8] Chapter 2 – Parachute Landing 2.1 Friction Constants In this chapter I will be using the details about the skydiver given in section 1.1. The subject of this chapter is parachute and friction amounts. The guiding question can be found below. This chapter will be guided by a set of questions and their answers (these can be found in paragraph 2.3). How long will it take to reach the ground with the following k values? k diver = 0.3 k parachute = 4 2.1.a Hypothesis I think that the time it will take to reach the ground for the skydiver (with the 100kg mass) will not be more than a few minutes or less than one, as the velocity at which the person would then travel the full length of 2000m would be 33m/s, or 120 kilometer per hour (the velocity of an average car on the highway). This is without using a parachute and will cause immediate death upon colliding with the ground or even beforehand (not being able to handle the deceleration that is a huge blow to the body). A source on the average velocity of a skydiver: “ The terminal velocity for a skydiver was found to be in a range from 53 m/s to 76 m/s. Four out of five sources stated a value between 53 m/s and 56 m/s. Principles of Phys- ics stated a value of 76 m/s. This value differed significantly from the others. Then again, the value is variable since the weight and the orientation of the falling body play significant roles in determining terminal velocity. [Speed of a skydiver] With our skydiver being a bit overweight to the most skydivers (100 kg is not the average weight for a human). See below for a more accurate source on weight. Table 1 - The average men’s weight Age 20 - 29 30 - 39 40 - 49 50 - 59 60 - 69 Kilograms 76 81.3 82.6 84 83.5 Average 81.84 [Men Weight] According to the source for men average weights, the average weight is 81,84. Therefore I can expect a slightly higher terminal velocity if my skydiver has a mass of 100kg . For example instead of a velocity of 53 m/s to 56 m/s, a velocity of 58 m/s to 60 m/s. [Page 9] 2.1.b Variables The variables in this experiment of finding out (terminal) velocity are: What do I change? - Independent Time taken - sec This is what I am trying to find out and will be visible in the graphs. It will be the intersection with the 2000 meter border and give a resultant duration of the flight from jump to finish. Therefore it is an independent variable (on the x-axis of a potential graph). The friction constant - the k in (formula for air friction) This will be inserted in two different formulas to fully simulate the skydiver's flight. This is also sensible because I am testing the whole flight with and without a parachute. If I would change this, my results would lose accuracy and consistency. What do I observe? - Dependent Distance Travelled - m This will define my time taken and is set at 2000 meters to calculate the time taken to reach this amount. The time will be my end calculation but not my observation here. The distance is set at the y axis in graph 5 on the next page. What do I keep the same? - Controlled The mass of the skydiver - kg This will be controlled in order to find accurate results. As I am looking for the result of one calculation this has to be a constant, single value and thus be controlled. I have however speculated with this to improve the accuracy of my assumptions. The Gravitational Force - N This is proportional to the mass and will therefore also be controlled if mass is con- trolled. 2.1.c Plan In this lab simulation I will try to find out the time it takes to land from jump to landing of a 2000 me- ter skydive. I will start with a scheme containing all my simulation calculations and notes guiding them. This is followed by the result readable from this scheme. And concluded by the relevant questions to work out the simulation in more depth. 2.2 Conducting the experiment and calculations See the next page for the full, illustrated and explained scheme. This is also figure 3. [Page 10] [Page 11] From the scheme I can conclude that the skydiver will have reached the ground after exactly 62 se- conds, a bit more than one minute. An additional fact derived from the scheme: the terminal velocity maximum is exactly 57,17 m/s and the minimum is 15,10 m/s. This is partially correct with my hypothesis: The terminal velocity of the skydiver is indeed around 58 m/s in the flight without parachute and around 15 m/s with parachute. This proves that this is caused by the small deviation from average weight and average measurements. 2.3 Questions Here I will state all the given questions and their elaborate answers. 1. Do you find the k parachute value of 4 kg/m big enough to have a safe landing? The value of 4 kg/m gives a terminal velocity of 15,10 m/s, which is 54,36 km/hour. A landing at this velocity (the velocity at which a car drives through populated villages or smaller roads) would be fatal and thus not safe. Therefore it should be changed: changing the k value to a range of values gives the results of table 2. Testing k values - Table 2 k value Terminal Velocity Terminal velocity convert- (m/s)(+/- ed to km/hour (multiplied 0,001m/s) by 3.6) 1 27,47 98,89 2 20,65 74,34 3 17,24 62,06 4 15,10 54,36 The velocity of 35,1 km/hour is 5 13,60 48,96 still too fast (the velocity of an 6 12,47 44,89 extremely fast bicycle). As the k 7 11,59 41,72 value is still decreasing yet with 8 10,87 39,13 decreasingly small amounts, I will 9 10,27 36,97 continue trying some higher k 10 9,75 35,1 values in table 3 below. Testing k values - Table 3 k value Terminal Velocity Terminal velocity convert- (m/s)(+/- ed to km/hour (multiplied 0,001m/s) by 3.6) 20 6,95 25,02 30 5,69 20,48 40 4,93 17,75 50 4,42 15,91 [Page 12] A velocity of around 16 km an hour is a fast bike and probably survivable. Therefore the k value should be more than 50 to have a completely safe and sound landing. 2. Find from the velocity-time diagram the deceleration of the diver after he opens his parachute. The best way to do it is to print a full screen velocity-time diagram and find the slope in the graph just after 30 seconds? To avoid making this lab report longer than necessary I will leave that out. Instead I will perform this on my computer screen. Below is an illustration. This means that the deceleration of the skydiver is -62,3 m/s in the first four-tenth of a second after the opening of the parachute. In other words, he is slowing down by -224,28 km/hour, which is simi- lar to being pushed back by an extremely fast driving vehicle (a force pushing him backwards with 224,28 km/hour. Even though the skydiver is not hit by a physical object he would still experience the exact same, which would be fatal to his body and kill him. [Page 13] 3. Calculate how many g’s he experiences (one g = 10 m/s2). Use your answer from ques- tion 2. The deceleration divided by 10 gives the amount of G’s. In this case this is an average of -6,23 G or 6,23 G (negative or positive does not matter here). See below: 4. Find out if the number of g’s can be handled by his body by using internet sources. Use a bibliography in your lab report.4 A human being can take about 3G before feeling anything. After an additional 2 or 3G they will feel sick and pass out. At 6G ever human would have lost consciousness. To be on the safe side I would suggest no more than 3 to 4 G exposal. However, our skydiver is experiencing more than 6G which would make him lose consciousness. Exposure to this any longer would kill the skydiver, it he had not [G-force] already died from this sharp G-force crushing him. 5. Do you think the simulation of the opening of the parachute is realistic enough? Explain by using the answers of the questions above. Give also comment on how to make the simulation better.5 I think this simulation comes reasonably close to realism. I am not sure about it though, but I always thought a 2000 meter skydive would take more than a few minutes. I could of course be wrong and the rest of the calculations are proven and show their accuracy. The only way to improve realism would to compare this to practical and real tests or to perform smaller, real life tests using masses and air friction. In addition, the simulation lacks realism because the skydiver jumps completely ver- tical, while a parachute and body cannot stay still like that. A sloped descent would increase the re- quired distance travelled which would require an additional calculation and variable. The safe k value of about 50 kg/m is appropriate in my opinion, as the person weighing 100 kg he should have an about 2 meters wide parachute (or 2m2 parachute), instead of just 4. The fact that the parachute with a k value of 4 would be unsafe has been proven by calculating the deceleration and G forces. This shows that the parachute would probably crush or damage his body either by land- ing or opening the parachute. The k value of 50 would make the person survive the landing, but has a lethal deceleration. To see all the effects of both k values, see graph 5 below. 4 See the end of the report for the bibliography. 5 This is also my evaluation of this chapter [Page 14] Some ways to improve the simulation and project are stated below. Find more variables to change while testing and test on more values and statements. An ad- ditional, lighter skydiver with another sized parachute would have a different force, graph, timing and G-force whatsoever. Learning more about Vissim will improve students’ skill and give them the possibility to add factors and variables to the lab files. This might help to improve the fun-factor and quality of the project. The simulation could have been improved with different heights instead of only 2000. Do watch out to not add too much extras or too repetitive material to avoid bored students that reluctantly work resulting in lack of quality work. More practice with k. It was given in this experiment so I didn’t come up with initial values or really looked at how they were calculated and why. This project was overall a fun and educative experience. The including of a real-life theme and exam- ple really gave it an interesting side and probably made others as enthusiastic about it as I was. I will [Page 15] probably have made some mistakes and miscalculations yet I managed to get Vissim working quite well and was introduced to this new kind of physical science in a great way. Bibliography Printed Sources Physics for You. Johnson, Keith. 2006, Physics for You - Revised edition for All GCSE Examinations, United Kingdom, Nelson Thornes Ltd. Electronic Sources The first word refers to in-text references in the text. Skydiver, Upload: Wikimedia - Skydiver [Online Image] 29 September 2008. URL <http:// upload.wikimedia.org/wikipedia/commons/9/9d/Skydiver.jpg> Speed of a skydiver, Speed of a Skydiver (Terminal Velocity). [Online] 30 October 2008. URL <http:// hypertextbook.com/facts/JianHuang.shtml> Men Weight, Men's Weight Chart [Online] 30 October 2008. URL < http://www.halls.md/chart/men- weight-w.htm> G-force, G-kracht - Wikipedia [Online] 30 October 2008. URL<http://nl.wikipedia.org/wiki/G-kracht> G-force, Wiki Answers - How many g's can a human take before they blackout? [Online] 30 October 2008. URL<http://wiki.answers.com/Q/How_many_g's_can_a_human_take_before_they_ blackout> [Page 16]