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Department of Mechanical and Industrial Engineering ME 3455: Dynamics and Vibrations (Spring 2023) Trajectories with Aerodynamic Drag Submitted by Gili Fang, Taylor Leibig, Dominic Natale, & Declan Rogers Abstract In an effort to further explore drag and its effect on projectiles an experiment with four trials was conducted using a tennis ball, iphone camera, tracker software, and matlab code. A tennis ball was thrown and filmed in slow motion with an iphone, then analyzed using tracker software. The data collected from the tracker software was imported into matlab to find results and plot them. On average the tennis ball traveled 2.475 ft with a velocity of 10.03 ft/s. The average experimental drag coefficient was .4967. Date Submitted: 2/19/2023 Date Performed: 1/30/2023 Lab Section: 30432 Course Instructor: Jahir Pabon Lab TA: Zahra Karimi

List of Symbols Name Symbol Terminal Velocity 𝑣 ? Initial Velocity 𝑣 0 Angle θ Acceleration due to gravity 𝑔 Height ℎ Distance traveled ? Drag coefficient 𝐶 𝑑 Density of fluid ρ Average velocity 𝑉 Cross sectional area 𝐴 Table 1- List of symbols Introduction Drag is generated by the interaction between a body and a fluid. Friction between the molecules of air and the surface of the moving object create a force in the opposite direction of the object’s motion. The magnitude of the drag force is dependent on aspects of the system such as the density of the fluid traveled through, the size and shape of the body, and the velocity the object travels with, however, its direction is always the opposite of the direction of velocity. When it comes to projectile motion, drag has the ability to reduce the horizontal distance the projectile travels and reduce the range of velocity, when compared to a situation where drag is negligible. As velocity increases, the drag force increases and approaches the magnitude of the force of gravity. When it does reach this point, a terminal velocity is reached. This value is the

maximum velocity an object in free fall can reach when drag is present and occurs at the same time the acceleration of the trajectory is equal to zero. The purpose of this experiment is to simulate projectile motion in order to observe the effects of drag on an object in air. To do this, a tennis ball was thrown and its trajectory was recorded. After calculating the distance traveled, and initial conditions (velocity, angle of inclination, and height) it was possible to compare the results with other models and verify the effect of drag on the simulated trajectory. Theory and Methods Given an initial velocity V 0 , an initial height y 0 , and initial angle of trajectory 0 0 , we can use the equation: to find the horizontal distance of ∆? = ?𝑜𝑠(θ)𝑣 0 × 𝑣 0 ?𝑖𝑛(θ)+ (𝑣 0 ?𝑖𝑛(θ)) 2 +2𝑔ℎ 𝑔 the foam ball. Figure 1- Free Body Diagram of trajectory with Equations of Motion Using Newton’s Second law of , we can derive the equations of motion in the 𝐹 = 𝑚𝑎 trajectories in the x and y direction as seen above. The mass of an object on a free falling object will change how fast the object is accelerating towards the ground. The more massive the object is the more they are acted upon by the force of gravity causing the object to need a greater air resistance force to even it out. The cross sectional area of an object will impact the amount of air resistance resulting on that object.

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If the object has an increase in cross sectional area, then there will be an increase in the amount of air resistance. Using the equation we can calculate the analytical terminal velocity of 𝐹 ? =− ρ? 𝐷 𝐴𝑉|𝑉| 2 the foam ball to be 50.78 ft/s. If we dropped the ball vertically, we used the MATLAB code to calculate the time it would take for the foam ball to reach 99% of its terminal velocity to be 4.17 seconds. Also using MATLAB, we were able to find the minimum height you would need to drop the ball for it to reach 99% of its terminal velocity to be 156 ft which you can see below. To begin our procedure we started by setting out smartphones to 240fps for a slow motion high speed video. Then we held the camera stable, perpendicular to the wall, to allow us to see the full travel of the foam ball in front of the pre-made grid without moving the camera. After we found a good place to keep the camera, we tossed the tennis ball at an arc and recorded the travel of the ball moving in an arc. We repeated tossing the ball 4 times to make sure we had sufficient data to use in MATLAB. Experimental Results Trial # 1 2 3 4 Initial Velocity (ft/s) 10.55 11.21 10.65 7.72 Initial Vertical Height (ft) 0.798 0.495 0.353 0.200 Initial Angle (°) 66.3 65.2 68.0 44.6 Horizontal Distance Traveled (ft) 2.299 2.780 1.785 3.036 Theoretical Horizontal Distance With Zero Drag (ft) 2.74 3.19 1.92 3.32 Table 2- Theoretical horizontal distance with no drag based on initial conditions

With an assumed drag coefficient of 0.47 and the initial conditions of each trial, a MATLAB program was used to determine the theoretical range of the thrown ball and landing time as shown in the figure below: Trial # 1 2 3 4 Simulated Range (ft) 2.3299 2.7901 1.7993 1.0455 Simulated Landing Time (s) 0.6940 0.6350 0.6600 0.4580 Table 3- Predicted distance with drag The trends seen in the following graphs show that the theoretical trajectory of a ball in air matches the experimental data observed. Each graph consistently shows two curves, one theoretical and one experimental, that are very close to each other. The second trial portrays the greatest distance traveled while the last graph gives the shortest distance. While all trials have very similar drag coefficients, the initial conditions of each trial are different. The second trial was released with the highest initial velocity in the x direction while the fourth trial was released with the lowest. These velocities are what propelled the ball in its reach and determined the distance traveled. The second trial also had a much larger initial angle than the fourth which allowed the ball to gain more height and distance before it fell. Using the initial conditions of each trial, a theoretical trajectory was mapped by modifying the drag coefficient to best imitate the arc of the experimental data. This new coefficient is shown for each trial and the mean of all four. Trial # 1 2 3 4 Mean Determined Drag Coefficient 0.4924 0.5033 0.4891 0.5019 0.496675 Table 4- Experimental drag coefficient Given these values, the standard deviation was determined to be 0.00699636.

Following the determination of an approximate experimental drag coefficient, the experimental data and a theoretical arc with this coefficient were graphed together for each trial as shown in the figures below. The determined coefficients proved to be quite accurate to the experimental data as the arcs for each trial are very close. Figure 2- Graph of trial 1 Figure 3- Graph of trial 2 Figure 4- Graph of trial 3 Figure 5- Graph of trial 4

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Discussion and Analysis This experiment was conducted under the effect of many potential sources of error. Since the trajectory of the ball was recorded in a room with both the air conditioning running and windows opened, wind in the room may have had an effect on the flight of the ball. Random error generated from angled positioning or slight movement of the camera affects all of the data since the videos are the basis for all conclusions. Additionally, the tracker software used to obtain the data of this experiment required very meticulous plotting of points on the trajectory of the ball that could easily be skewed from a view that is not zoomed in enough or incorrect due to camera movement during the fall. In the practical application of projectile motion within the game of baseball, wind conditions play a major role. Utilizing a MATLAB program, the necessary exit velocity of a home run at Fenway Park hit into left field was determined with the initial conditions as follows: Weight = 5.125 oz; Circumference = 9.125 in, Exit Angle = 45 degrees; Initial Height = 4 ft; Target Range = 345 ft; Target Height = 37 ft. Given negligible wind conditions, the required exit velocity was determined to be 174 ft/s. Given a 10 mph tailwind, the required exit velocity was determined to be 156 ft/s. Given a 10 mph headwind, the required exit velocity was determined to be 201 ft/s. Conclusions To summarize, the purpose of this experiment was to explore the effect of drag on projectiles. The experiment utilized a tennis ball, iphone camera, tracker software, and matlab code and consisted of four trials. On average the tennis ball traveled 2.475 ft with a velocity of 10.03 ft/s. The average experimental drag coefficient was .4967. The average theoretical distance without drag was 2.793 ft and the average experimental distance traveled was 2.475 ft. Without drag the tennis ball would have traveled an average of 0.318 additional feet. Acknowledgments Thank you to our TA Zahra Karimi for providing a thorough explanation of this lab at the beginning of class and assisting us with MATLAB issues through helpful insight and guidance.

References 1. S.S. Rao, Mechanical Vibrations, Fourth Edition, Prentice-Hall, Inc., New Jersey, 2004. Appendix A: Matlab Code global alpha; global m; global g; global tstep; global tend; global wind; Xo=0; % Initial Horizontal Position (ft) dia= 2.76; % Projectile Diameter (in) rho= 2.38e-3; % Air Density (slugs/ft^3) g= 32.2; % Acceleration Due to Gravity (ft/s^2) m= 1.98e-3; % Projectile Mass (slugs) C= 0.4924; tend=10; % Maximum time solved for. Increase this for more values (s) tstep=0.001; % Time Step (s) nu=(1.516*10^-5)*(3.281*3.281); % Kinematic Viscosity (ft^2/s) Yo= 0.798; Vx = 2.954; Vy = 10.125; Vo = sqrt(Vx^2 + Vy^2); %Initial Velocity (ft/s) T= atan2(Vy,Vx) * 180 /pi; %Initial Angle (degrees from horizontal) wind=0;

A=pi*(dia/(2*12))^2; % Cross-Sectional Area alpha=.5*rho*C*A; Re=Vo*(dia/12)/nu; % Reynolds number calculation ang=pi*T/180; t=0:tstep:tend;[t,X,Y,Vx,Vy]=Motioncalc(Xo,Yo,Vo,ang,C); for count=1:1:length(Y) if Y(count)<0 break; end end disp('Theoretical Zero-Drag Landing Time (s):');((Vo*sin(ang)+sqrt((Vo*sin(ang))^2+2*g*Yo))/g) disp('Simulated 2D Drag Landing Time (s):');t(count-1) disp('Simulated 2D Drag Landing Point (ft):');X(count-1) Exp_Data_File % (change) execute the file contains the experiemental data = [t, x, y] plot(X(1:count-1),Y(1:count-1) ,'DisplayName' ,'Theory'); % plot theoretical projectile with drag hold on plot( (data(:,2)), (data(:,3)+Yo) ); % plot the experiemenal projectile legend('Theory with drag', 'Experiement') title('Trajectory'); %Graph title xlabel('Distance (ft)'); %x-axis label ylabel('Height (ft)'); %y-axis label hold off % calculate mean squared error xv = linspace(max(X(1),data(1,2)), min(X(end),data(end,2)),200); ytheo = interp1(X,Y,xv,'pchip'); yexp = interp1(data(:,2), data(:,3)+Yo,xv,'pchip');

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disp('The MSE error between theory and experiment is:');mean((ytheo-yexp).^2) Vt=sqrt(m*g/alpha); [t,Xt,Yt,Vxt,Vyt]=Motioncalc(Xo,Yo,0,-pi/2,C); % Terminal Velocity Calculation for i=1:length(Vyt) if -Vyt(i)>=Vt*0.99 break end end plot(t,-Vyt(:)); xlabel('Time (s)'); % x-axis label ylabel('Velocity (ft/s)'); % y-axis label title('Terminal Velocity Simulation') disp('Analytical Terminal Velocity (ft/s):');Vt disp('Approx. Simulated Terminal Velocity (ft/s):');-Vyt(i) disp('Time to 99% of Terminal Velocity (s):');t(i) disp('Minimum Drop Height for Terminal Velocity (ft):'); -Yt(i)+Yo plot(-Yt,-Vyt);xlabel('Drop Distance (ft)'); % x-axis label ylabel('Velocity (ft/s)'); % y-axis label Yo=4; %Initial Vertical Position (ft) dia=9.125/pi; %Projectile Diameter (in) T=40; %Initial Angle (degrees from horizontal) m=5.125/514.785; %Projectile Mass (slugs) ang=pi*T/180; tend=20; % A=pi*(dia/(2*12))^2;

alpha=.5*rho*C*A; disp('Initial Velocity Without Wind (ft/s):');wind=0;Baseball(Xo,Yo,ang,C) disp('Initial Velocity With -10 Wind (ft/s):');wind=10;Baseball(Xo,Yo,ang,C) disp('Initial Velocity With +10 Wind (ft/s):');wind=-10;Baseball(Xo,Yo,ang,C) function Vo=Baseball(Xo,Yo,ang,C) THERE=1; for Vo=150:1:250 [t,X,Y,Vx,Vy]=Motioncalc(Xo,Yo,Vo,ang,C); for count=1:length(Y) if Y(count)>37 THERE=count; end end if X(THERE)>344 && X(THERE)<346 break end end %THERE %X(THERE) end function [t,X,Y,Vx,Vy]=Motioncalc(Xo,Yo,Vo,ang,C) %Define variables global alpha; global m; global g; global tstep; global tend;

global wind; t=0:tstep:tend; Vox=Vo*cos(ang); Voy=Vo*sin(ang); wind2=wind*1.467; % mph to ft/s conversion X=zeros(length(t),1);Y=zeros(length(t),1); Vx=zeros(length(t),1);Vy=zeros(length(t),1); Ax=zeros(length(t),1);Ay=zeros(length(t),1); X(1)=Xo;Y(1)=Yo;Vx(1)=Vox;Vy(1)=Voy; % Equations of motion with initial conditions Ax(1)=-alpha/m*((Vox+wind2)^2+Voy^2)^0.5*(Vox+wind2); Ay(1)=-g-alpha/m*((Vox+wind2)^2+Voy^2)^0.5*Voy; % Loop to numerically solve non-linear differential equation using a % variation of the central difference approximation method for i=1:length(t)-1 % Update velocities using forward differences Vx(i+1)=Vx(i)+Ax(i)*(t(i+1)-t(i)); Vy(i+1)=Vy(i)+Ay(i)*(t(i+1)-t(i)); % Update positions using the average of the speeds between previous % and new values (for better accuracy) X(i+1)=X(i)+1/2*(Vx(i)+Vx(i+1))*(t(i+1)-t(i)); Y(i+1)=Y(i)+1/2*(Vy(i)+Vy(i+1))*(t(i+1)-t(i));

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% Update accelerations (equation 1 in the lab handout) for the next % step using the new speed values Ax(i+1)=-alpha/m*((Vx(i+1)+wind2)^2+Vy(i+1)^2)^0.5*(Vx(i+1)+wind2); Ay(i+1)=-g-alpha/m*((Vx(i+1)+wind2)^2+Vy(i+1)^2)^0.5*Vy(i+1); end end Appendix B: Data Table Trial Number Initial x velocity (Vx) Initial y velocity (Vy) Initial height (Yo) Horizontal distance without drag Horizontal distance with drag Experimental Horizontal Distance 1 2.954 10.125 .798 2.74 2.3299 2.299 2 4.484 10.275 .495 3.19 2.7901 2.780 3 2.699 10.301 .353 1.92 1.7993 1.785 4 2.149 7.416 .200 3.32 1.0455 3.036 Appendix C: Teamwork Breakdown Taylor: Tracker Software, List of Symbols, Conclusion, Abstract, Post Lab #4, 5 Declan: Post Lab #6, 8, 9, 11, 12, 13 Dom: Theory and Methods, Pre lab 1, Post Lab #1, 2, 3 Gili: Introduction, Pre Lab #2, Post Lab #7, 10, Acknowledgements