Moving away from Happy Physics World

Transcription

1 Moving away from Happy Physics World Inquiry 4 William Oakley Abstract Introduction As a former Mechanical Engineer, I ve had a great fondness for Newtonian Mechanics, to the point that I m getting certified to teach Physics, Chemistry and Engineering in addition to Math. One thing that has bugged me about Newtonian Mechanics, and physics in general, is that everything happens in ideal conditions, or as my Pre-AP Physics Teacher described it, Happy Physics World (HPW). HPW is a magical land where air resistance doesn t exist, cars are able to maintain constant acceleration on roads with no traffic and stoplights, friction is arbitrary and rolling objects always invoke the no-slip condition. In all seriousness, the ideal conditions that most physics operates under makes me wonder just how useful they are in real life. For example, projectile motion makes the assumption that air resistance does not exist. So what better way to investigate this than a full scale Inquiry? Background Projectile motion is defined by the kinematic equations of linear motion when it is first taught to students without a calculus background. While the equations have been verified by nearly every high school physics classroom at some point, Newton invented an entire branch of mathematics to create them. To compromise between the two methods of proving the kinematics stated previously, I will use Calculus, Newton s Laws of Motion and using the definitions of Position S(t), Velocity V(t) and Acceleration A(t) as functions of time and with each other. Let s assume that we don t know what the acceleration of gravity is, but we can make the assumption that it is constant and later test the final derivations to see if our assumption is accurate. What we do know from rearranging the equation that whatever total acceleration acts upon the ball is equal to A = F M. With the constant established, we can then use implicit differentiation on the definition of acceleration as such: A = dv dt, A dt = dv, A t + C = V(t). If we set the time equal to zero, then the acceleration cancels out and what is left is whatever the velocity is at the start. Thus, the constant is equal to V 0 and thus gives us the first Kinematic Equation of: A t + V = V(t) (1)

2 Using implicit differentiation on our new equation produces the following (substituting velocity with definition): A t + V = ds dt A t + V = ds dt, (t A )/2 + t V + C = S(t). If we apply the same logic as before, setting t = to zero sets all other terms to zero by multiplication except for C, thus C represents the initial position. Thus the second equation becomes: (t A )/2 + t V + S = S(t). (2) These two equations are all that is needed to analyze projectile motion. There are technically 4 kinematic equations of constant acceleration that are taught to students, but the other two are merely re-arrangements of these two equations, thus the full set looks like this: Figure 1- The 4 basic kinematic equations. Numbering in image is not used in paper. Using empirical values, or the values from my own experiment, we can determine the acceleration of gravity, which is defined as 9.81 m/s 2 in the y direction, and will be referred to as g for the rest of the inquiry. (negative depends on orientation of vector axis). (To Dr. Mardar, if you wish me to use this as my mathematical modeling section, please advise). Now, comes the hard part: trying to account for air resistance with the Newtonian equations. Fortunately the course book has a section that specifically uses dimensional analysis to determine the frictional force on a ball in projectile motion due to air resistance, which is exactly what we needed. In an effort to save space, F =.5 C A ρ v will be the starting point of the mathematical modeling (also known as the drag equation). C represents the Drag Coefficient, A represents the cross sectional area facing the force, rho the density of the air and finally v for the velocity. For convenience sake, let us reduce all constants to the following: D = (ρ C A)/2. (3) We are also making the assumption that the air is still, so the force vector acts in the exact opposite direction of the velocity vector, which means that the friction force will act in both the x and the y direction (and also that the velocity acts relative to the air as it does to the ground. Before continuing, we need to specify that the above equation is strictly the magnitude of the friction force, where V = V + V. We still need the unit vector component for the force before we can analyze it with the Newtonian equations (and apply it to a FBD). With the above assumption of still air, we can write out Ff = D v v whose components consist of the following: Ff = D v v

3 Ff = D v v. Now that we have both magnitude and direction, we can create a free body diagram: we Figure 2- Free Body Diagram of marble in launch As figure 1 represents a graphical interpretation of the forces acting on the ball in the x and y directions, we can continue algebraically: F = Ff and F = Ff m g. Since we are only concerned with acceleration, we can divide by mass in both the x and y directions to remove all mention of Force from the equations, leaving us with: Ax(t) = D m V and Ay(t) = D m V g. Using previous notations established for V, we can transform the above equations into differential equations. d x(t) dt = ( ) and d y(t) dt = ( ) g And now we have two nonlinear differential equations which requires a computer in order to solve. Fortunately the problem can still be solved numerically by making the assumption that the over a small enough interval of time we can come approximate the solution by applying previously established kinematic equations. Using what we have established, we can define a new set of equations based on the old ones with drag applied by using the same process. First we start with a new definition of acceleration for both the x and y directions using the differential equations previously established (this also makes the assumption that

4 the overall acceleration is. To reduce further clutter, let us combine the D/M into a new constant, J. We now have the following: a = J v v and a = J v v g (4 and 5) Using the incremental method, we can determine the velocity as such: v + v = v + a t and v + v = v + a t. (6 and7) Finally, by the same reasoning as before, the equation for position becomes: x + x = x + v t + a ( t ).5 and y + y = y + v t + a ( t ).5 (8 and 9) Now that an incremental formula has been determined, we can apply it in a computer program like excel, which will be discussed in the Data section. Experimental Design The experiment requires a video camera that is capable of recording the entire launch from start to the first landing point, so before starting make sure that there is enough room The experiment needs to be conducted in a vacant area to avoid hitting anything anyone. For extra safety, safety goggles should be worn at all times. There are many devices that can be used to perform projectile motion like catapults, trebuchets, air-pressure cannons, etc. This experiment uses a spring powered marble launcher (specifically this one: Launcher.html ) because the spring driven system is simple enough to control and won t launch the marble too fast or too far away for the camera to pick up. Depending on the quality of your camera, it may be hard to see the marble. It is best to use a bright colored marble set against a black background. Black poster board or butcher paper will work perfectly. When setting up the launcher, pick a spot with an easily identifiable marker to make sure the Launcher is not moved after each firing. This will also be needed for video analysis with Logger Pro. It is also suggested to have some sort of catch fence to help stop the ball from rolling away, as you only need the initial point of contact for analysis. It is suggested to have the launcher, camera and landing area at generally the same height to make it easier to record the launch. Once everything is set up to record the launch, it s time to begin the experiment. The marble will be launched at 30, 45 and 60 degrees from the horizontal (ground/table) 5 times (at least), for a total of 15 trials total. After recording the video footage, the next step is to use Logger Pro to start analyzing the videos. Once you insert the video, position the slider at the first frame where you see the marble launched. Then, use the scale button on the meter stick along it s full length to give the program a reference for measuring distance. Establish a coordinate plane with the axis tool, preferably lining up the origin for convenience. Finally, using the point tool, we can track the path of the projectile by in each frame with a point. If blurs

5 occur, use a consistent system for marking. This Inquiry uses the front of the blur as the established point. The system will record time, X and Y coordinates. I copy and pasted these into Excel to perform the rest of the analysis, which is explained in greater detail in the Data section. Data (Due to the fact that video was used as the only source of evidence with Logger Pro, there may be some inconsistencies between trials on where the start and stop locations as the origin must be manually established. In addition, all data coordinates were input by hand, using the very front of the ball blur as the established point. Finally, the 60 degrees videos do not include the very top of the arc (the ball went off screen), so for the 60 degree treatment, the Ymax value will not be calculated or included in the analysis) For the analysis, the following data values are taken from the Logger Pro for teach trial: Xinitial, Yinitial, Xmax, Ymax, Total Time (from launch to first landing), Vx Intial, Vy initial, Vx final and Vy final. To compare the accuracy of the kinematic equations with and without air resistance, an actual set is needed for comparison. After the data is uploaded into excel, the above variables were calculated and are displayed below. Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 Average St. Dev. Xi (m) Yi (m) Xf (m) Yf (m) T (s) Vix (m/s) Viy (m/s) Table 1: Collected data from 30 degree treatment Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 Average St. Dev. Xi (m) Yi (m) Xf (m) Yf (m) T (s) Vix (m/s) Viy (m/s) Table 2: Collected data from 45 degree treatment

6 Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 Average St. Dev. Xi (m) Yi (m) Xf (m) Yf (m) T (s) Vix (m/s) Viy (m/s) Table 3: Collected data from 60 degree treatment In addition to this data, we need to determine the values of D and k as previously defined in Background Section. Fortunately all the necessary data has been measured and is displayed in the below. The Drag Coefficient was taken from it s shape as a sphere, the mass and area were measured by hand and the Air Density was taken from a table assuming room temperature. Air Density Coefficient Mass(kg) Area(m^3) D J Table 4: Constants needed for Analysis Analysis In order to determine the validity of the kinematic equations with/without force, we need to determine the values of the experimental data, which will act as the control for the analysis. Because the data being used in the comparison comes from the same source (all input variables), we can use the one sample t-test, using the results from the experimental data as the population mean. As such, we use the following hypothesis: H 0 = There is no significant difference between the sample mean and the population mean. H a = There is a significant difference between the sample mean and the population mean. With alpha =.05, we can now analyze the data. The first test investigates what happens when calculating the final y velocity of the object. The experimental data is calculated by taking the last two points in the video and applying the equation V =. For the kinematic equation without air resistance Eq. 1 with t set to the maximum elapsed time and all constants applied.

7 To calculate with air resistance, Equation 7 was entered in excel with the initial conditions already defined, and then applied over and over for all subsequent velocity values, using the average elapsed time and a time step difference of.04 for the duration. Standard Deviations in data was used from the combination of the time and initial velocity average No Air Air Actual Table 5: Final Velocity calculation results, mean values Running a simple t-test, we obtain the following results: No-Air Air Table 6: P-Values for final velocity Given our initial hypothesis, none of the data s p-values are below alpha, so all equations do not statistically significantly differ from the actual. However, the p-values are somewhat over the place, so I m not sure how accurate of an analysis this really is. Conclusion *Note to Dr Mardar- I am unsure if the current analysis is correct, so I am refraining from concluding until we can talk about the analysis section.

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