This year, the university physics competition posed a question on how the diameter of a table tennis ball affects the trajectory of a table tennis ball. After 48 hours, Jeffrey Heninger, Nathan Holley, and Aishwarye Chauhan produced the following paper in response.For more information on the problem and competition, see http://www.uphysicsc.com/.
Team 458 1 How the Trajectory of a Table Tennis Ball is determined by its Diameter Team 458: Problem B Abstract We are investigating how changes in the diameter of an official table tennis ball for international competition will affect its trajectory. Our answer to the question of which diameter is “best” approaches the problem from the perspective of the general television viewing audience. We accomplish this by running several simulations based on gathered statistics of table tennis equipment, human physiology, and previous research to find the diameter with the trajectory that matches the requirements for an ideal table tennis game. Two main components influence our assumptions for the best diameter: visibility and speed of game play. We assume that there is some angular velocity relative to the camera, over which the general audience can no longer comfortably and consistently observe the ball, and thus the game. The best game maximizes the speed of the ball, which maximizes speed of game play and the challenge of the game, while still staying under this observable limit. From our data, we agree with the ITTF’s decision to increase the ball size from 38 mm to 40mm, and deem that 41 mm to be diameter that maximizes both visibility as well as average game speed. Any diameters larger than this limit are unnecessarily slowing the game down, as well as detracting from angular momentum, or the spin, of the ball.
Transcript
Team 458 1
How the Trajectory of a Table Tennis Ball is determined by its Diameter
Team 458: Problem B
Abstract
We are investigating how changes in the diameter of an official table tennis ball for international competition will affect its trajectory. Our answer to the question of which diameter is “best” approaches the problem from the perspective of the general television viewing audience. We accomplish this by running several simulations based on gathered statistics of table tennis equipment, human physiology, and previous research to find the diameter with the trajectory that matches the requirements for an ideal table tennis game.
Two main components influence our assumptions for the best diameter: visibility and speed of game play. We assume that there is some angular velocity relative to the camera, over which the general audience can no longer comfortably and consistently observe the ball, and thus the game. The best game maximizes the speed of the ball, which maximizes speed of game play and the challenge of the game, while still staying under this observable limit.
From our data, we agree with the ITTF’s decision to increase the ball size from 38 mm to 40mm, and deem that 41 mm to be diameter that maximizes both visibility as well as average game speed. Any diameters larger than this limit are unnecessarily slowing the game down, as well as detracting from angular momentum, or the spin, of the ball.
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Introduction
In 2000, the official diameter of the table tennis ball for international competition was changed from 38 mm to 40 mm. This decision was made in order to make the sport more entertaining, by slowing the ball down. Certain questions are raised, such as, “What about this decision makes the sport more entertaining?”, and “Is this the optimum size?”
A slower ball means two things: that the ball in movement is easier to follow, and that the ball is easier to keep in play, based on human reaction times. We investigate this phenomenon with the goal to optimize the time of game play in which the average audience can comfortably observe the ball. We also want to keep the speed of the ball on the order of human reflexes, to avoid making the game too easy or slow-paced.
Court Set Up
To begin investigating this problem, we first have to establish the basic setup of the table tennis court. According to the official regulations of the International Table Tennis Federation (ITTF), the table must be 2.74 meters long, 1.525 meters wide, and 0.76 meters tall.[1] We choose to set up a right-handed coordinate system with the origin located at the center of the table, with the x-axis extending down the length of the table, the y-axis extending along the width of the table, and the z-axis pointing upwards. The net extends along the y-axis to a height of 0.1525 meters.
The players in a table tennis tournament stand back from the table and serve from above the height of the table. To account for this, we assumed that the players strike the ball from 0.7 meters above the table and 0.63 meters back from the table, 2 meters away from the net.
Simulation of the Trajectories
Our simulation focuses on the trajectory of the ball as it travels between players. We choose not to consider the serving strokes, which are highly variable and are easily manipulated by the server, but rather to focus on the strokes during the volley. These trajectories must pass over the net and strike the opposite side of the table before they are returned.
We assumed that the player gives a 40 mm ball an initial velocity, v0, corresponding to 20 m/s in the x-direction and 4 m/s in the negative z-direction. These values were chosen as the result of an interplay of several factors. First, we wanted to make sure that these speeds are reasonable values for table tennis competition. In a competitive tournament, the ball traverses the length of the table approximately four times per second. [2] Second, the ball must clear the net and strike on the opposite side of the table. By adjusting the initial velocities, we were able to fulfill both of these conditions, even when we subsequently varied the initial velocity in the y-direction, the spin of the ball, and the diameter of the ball.
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The player also imparts an initial spin,ω, on the ball. We allowed each component of the angular velocity to vary from – 20 rad/s to 20 rad/s independently. These values of angular velocity are within the range the ITTF estimates for table tennis ball spin and ensure the ball remains above the net and on the table.[3] We also assumed that the angular velocity does not vary appreciably over the trajectory, as has been previously justified in the literature.[4]
Forces Acting on the Ball
Aside from the initial driving force from the paddle, the three individual forces acting on our ball at any given time are gravity, drag, and the Magnus force.
A constant gravitational force, Equation (1), is acting on our ball in the downward direction and is the result of the attraction between the Earth and the ball.
FG=−m g z (1)
The Drag force, Equation (2), is acting in the direction opposite to the motion of our ball. It is a resistive force and is proportional to the square of the velocity. It depends on:
1. The density of the fluid in which the ball is traversing. For air, ρ = 1.225 kg/m3. [5]
2. The projection of the apparent surface area on a plane perpendicular to ball’s motion, A = π r2.
3. And, an experimental constant known as the drag coefficient that is used to quantify the drag or resistance of an object in a fluid environment. For a sphere, Cd = 0.47.[6]
FD=−12
ρ A Cd‖v‖v [4] (2)
The last force in our force equation is the Magnus force, Equation (3). An object spinning in a fluid experiences this force orthogonal to its line of motion. If the angular velocity is high enough, this force produces a significant change in the trajectory of the ball. The curvilinear motion of the ball in a ping pong game is attributed to this force.
FM=2 π r3 ω× v [4] (3)
Thus, the entire equation of motion becomes:
v '=∑ Fm
=−g z− 12 m
ρ A Cd‖v‖v+ 2m
π r3
ω× v(4)
The ball also undergoes an inelastic collision when it hits the table. We assume that the friction on the table is small, so the only change in the velocity as a result of the collision is in the z-direction. The ball loses some energy as a result of the inelasticity of the collision. This loss of energy is determined by the coefficient of restitution, ε = 0.9. [7] The ball then continues on its trajectory with the new initial speed. We assumed that the spin is not significantly changed as a result of this collision.
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Since this is a three dimensional system of nonlinear ordinary differential equations for v, we do not expect an analytic solution to exist. Instead, we use a fourth order Runge-Kutta method in MATLAB to approximate the solution, subject to the initial conditions listed above and the boundary condition corresponding to the inelastic collision when z=0. To calculate the trajectories of the ball, we integrate the resulting v (t) with respect to time.
Figure 1: The trajectory of the ball for varying spin. Each component of the angular velocity is allowed is be - 20 rad/s, 0 rad/s, or 20 rad/s. The 27 resulting trajectories are shown. All of the trajectories are above the net at x = 0 m and strike z = 0 within the dimensions of the table.
Figure 2: The trajectory of the ball for varying initial velocities. The initial x-velocity is allowed to be 19 m/s, 20 m/s, or 21 m/s; the initial y-velocity is allowed to be – 1.5 m/s, 0 m/s, or 1.5m/s; the initial z-velocity is allowed to be – 3.5 m/s, - 4 m/s, or – 4.5 m/s. All the trajectories clear the net and hit the table.
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Diameter Dependence
Ultimately, the problem is related to how the sport of table tennis depends on the diameter of the ball. Some of the effects of changing the diameter are readily apparent in the equations of motion. Other effects are less direct. They result from the change in mass and moment of inertia and the associated initial velocity and angular velocity that accompany the change in diameter.
Our first task is to calculate how the mass changes with the diameter. We know the mass for the 38 mm and the 40 mm balls to be 2.5 grams and 2.7 grams respectively, as stated by the ITTF.[3] Since the table tennis ball is a hollow spherical shell, we do not immediately know the relationship between mass, M and diameter, D. We can use our two data points, along with the limit that a diameter of 0 mm corresponds to a mass of 0 grams, to calculate our mass dependence on diameter. We use a generic power law,
M (D )=κ∗D γ (5)where κ is some constant and γ is our power. We solve this equation to find that κ is equal to 0.3375 kg/m3/2, and γ is equal to 3/2.
This change in mass affects how hard the players can hit the ball. If we assume the player imparts a specific impulse to the ball on each stroke, the increased mass will correspond to a decreased initial velocity. This is easy to quantify: ∆ p=m v0 is constant, so
m v0=M V 0
⇒V 0=
mM
v0(6)
The mass change similarly affects the angular velocity of the ball. The moment of inertia
of a hollow sphere is given by i=23
m r2, where r is the radius of the ball. We assume that the
player imparts an initial angular impulse to ball on each stroke. The new angular velocity will be given by:
23
m r2ω=i ω=I Ω=23
M R2 Ω⇒
Ω= mr2
M R2 ω(7)
These changes to the initial velocity and angular velocity are the most pronounced effects of the change in diameter.
Maximum Visible Speed
There are two opposing forces that determine how interesting table tennis is to watch. The game should be fast, so the game play is exciting and both players are always operating at the limit of their reaction time. However, the game should not become too fast that it is impossible for spectators to follow the trajectory of the ball. We want to choose the diameter of
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the ball so that it maximizes the speed of the game, while still remaining visible to the audience throughout of the game.
To determine this, we first have to determine the speed at which a TV spectator can no longer follow the ball. We define easy to follow as smooth pursuit – the motion of the eye continuously following a moving object. The maximum angular velocity that the average human eye can follow using smooth pursuit is 90.9 °/s [8]. Humans can follow faster moving objects using catch-up saccades; however, this makes the play much more difficult to follow.
To determine the maximum velocity of the ball the human eye can follow, we need a relationship between the angular velocity seen on TV and the actual motion of the ball. We assume that the players stand 4 meters apart and that the TV camera is a distance, d, 13 meters away, in accordance with the ITTF’s recommendations.[9] This configuration is shown in Figure 3. This camera distance is far enough away that the distance to the ball does not change appreciably, so we can relate the x-component of the velocity to the angular motion our eyes see. Using this information, we determine
θ=2 tan−1( L/2d ) (8)
V max=L∗θmax
θ(9)
This gives us a maximum speed of 19.2 m/s. Above this speed, the audience will be unable to clearly resolve the ball, and thus unable to follow the game.
Figure 3: The TV camera set up for a C
2-2
θ
-1 10
d
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table tennis tournament. The camera is located at point C, a distance d from the table. The game takes up an angular distance θ relative to the camera. Figure is not drawn to scale.
Results
The key question we are hoping to answer is what percent of the time a TV spectator can view the ball during each trajectory. To accomplish this, we record two times for each trajectory: the time its x-velocity first drops below the maximum visible speed and the time the ball reaches the opposite player. We then take the ratio of these two times.
We allow the value of the diameter to change between the values of 30 mm and 48 mm. This change influences the equation of motion through several mechanisms. The radius of the ball directly appears in both the drag and Magnus forces. The change in diameter also corresponds to a change in mass, which further affects the acceleration due to these two forces. The change in mass also affects the initial velocity and angular velocity the players impart to the ball.
Rather than just relying on one trajectory to determine the ratio of these times, we allow the values of each component of angular velocity to vary from - 20 rad/s to 20 rad/s. Figure 4 and Figure 5 show how the x-velocity depends on time for these trajectories.
Figure 4: The x-velocity vs. time as the diameter of the ball varies from 30 mm to 38 mm. The different curves for each color correspond to the different spins of the ball. The balls reach the other player between 0.15 sec and 0.25 sec, depending on the trajectory.
Legend: 38 mm 32 mm36 mm 30 mm34 mmMaximum Visible Speed
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Figure 5: The x-velocity vs. time as the diameter of the ball varies from 38 mm to 48 mm. The different curves for each color correspond to the different spins of the ball. Notice that the balls larger than 41 mm never exceed the maximum visible speed. The balls reach the other player between 0.25 sec and
0.3 sec, depending on the trajectory.
To get an accurate approximation of the percent of time one can see the ball with a given diameter, we averaged the results of the different trajectories. The graph of this average percent of time the ball is easily visible to a TV audience vs. the diameter of the ball between 30 mm and 48 mm is shown in Figure 6.
Several interesting results can be seen in the graph. The 38 mm ball that had previously been used by the ITTF can only be seen 82.5% of the time during typical game play. By changing to a 40 mm ball, the ITTF allowed TV audiences to see the ball 94.5% of the time. Any ball with diameter above 41 mm can always be seen during typical game play.
Legend: 38 mm 44 mm40 mm 46 mm42 mm 48 mmMaximum Visible Speed
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Figure 6: The percent of time that the ball is visible vs. the diameter of the ball. Balls 41 mm in diameter or larger are always visible during typical game play.Discussion
This approach gives us the ability to test the fundamental forces and trajectories acting on balls of varying diameters, in a very stable and consistent test. These major effects on the ball should present us with a good basis for which to predict the action of these balls. It is not difficult to look at our trajectories and note that they match what we expect for a table tennis ball to follow. We can also compare our numbers with various other measurements in literature and show that our numbers are good estimations, and have strong correlation to other reported data.[2]
[3]
There are some weaknesses to our approach, which are mostly due to differences between our approximations and real table tennis. In reality, the initial conditions for the ball are going to vary wildly, and are subject to the whim of the player. We are only modeling one very simple stroke here. In a real table tennis game, we cannot begin to take into account all the possibilities; however, since we are testing the balls only against each other, we can still assume that our data has a very real effect on the game play, and that general trends will be applicable. We are also basing our conclusions on what we are calling the “best diameter”, based on visibility of the ball and speed of normal game play. Obviously this is highly subjective and can vary heavily on the choice of initial conditions. We are assuming that the ball will travel in a random range of trajectories more or less about our initial conditions. We are not assuming that with our chosen diameter, the audience will always be able to follow the ball, but that with our diameter, the ball
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will tend towards speeds in which the average person can observe the ball at most times and normal game play is still fast paced.
Conclusion
The primary question in this problem is what makes a table tennis tournament fun to watch. There are two primary factors playing off against each other: the game must be fast enough to be exciting and challenging for the players, yet slow enough for a televised audience to be able to follow the game. Ultimately, the ITTF must decide how these two factors interplay to create the best spectator experience.
We can give the ITTF the information they need to be able to fully weigh the costs and benefits of changing the ball diameter. Figure 6 shows how the diameter of the ball determines the percent of time the human eye can follow the trajectory of the ball during normal game play.
We believe that TV spectators should always be able to follow the ball during normal game play. The only time the ball should move too fast to see is during a spike, or other exceptional speeds. To accomplish this, we conclude that the optimum ball is diameter 41 mm. Any increases in size beyond 41 mm will unnecessarily slow the game down.
[2] M. Alexander, A. Honish, http://umanitoba.ca/faculties/kinrec/research/media/table_tennis.pdf
[3] Y. IImoto, K. Yoshido and N. Yuza, http://www.ittf.com/ittf_science/SSCenter/docs/200200023%20-%20Llmoto%20-%20Rebound.pdf
[4] Dupeux, Guillaume, Anne Le Goff, David Quéré, and Christophe Clanet. "The Spinning Ball Spiral." New Journal of Physics 12.9 (2010). http://iopscience.iop.org/1367-2630/12/9/093004/pdf/1367-2630_12_9_093004.pdf
function r_dependence % calculates the average x velocity for a given radius clc clear close all DVec = .03:.0001:.048; frre = length(DVec); SeeTimeVec = zeros(1,frre); for ndx = 1:length(DVec) D = DVec(ndx); % diameter of ball [~, ~, ~, SeeTime, ~, ~] = runge_kutta_drag_eq_solve_w(D); SeeTimeVec(ndx) = SeeTime; end DVEC = 1000.*DVec; SEETIMEVEC = 100.*SeeTimeVec; plot(DVEC,SEETIMEVEC); grid on; axis equal; xlabel('diameter in mm'); ylabel('percent of time you can see the ball');end
function [XVal, YVal, Time, Height, AvVelocityX] = runge_kutta_drag_eq_solve_vy% Solve the diff eq for a ball moving through air clear clc close all global k g K t_step t_max eps color maxVx color = 'b'; % color of the graph g = 9.8065; % gravitational acceleration in m/s^2 rho = 1.225; % density of air in kg/m^3 c_d = .47; % drag coefficient of a sphere eps = .9; % coefficient of restitution for the ball maxVx = 19.21; % maximum speed clearly visible m40 = .0027; % mass of 40mm ball Rad40 = .02; % radius of 40mm ball D = .04; % diameter of ball Rad = D/2; % radius of ball m = .337493 * D .^ 1.5; % mass in terms of D % calculated using m and D for 38mm and 40mm ball A = pi*Rad^2; % cross-sectional area of ball in m^2
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k = .5 * rho * c_d * A / m; K = 2*pi * rho * Rad^3 / m; v = m40/m .* [20;0;-4]; % initial velocity in m/s r = [-2;0;.7]; % initial positions in m W = [0;0;0]; % initial spin t_step = .001; t_max = .5; XVal = zeros(3,3,3); % x value of bounce YVal = zeros(3,3,3); % y value of bounce Time = zeros(3,3,3); % time to reach the other player Height = zeros(3,3,3); % height at net possvx = m40/m .* [19,20,21]; % possible values for vx possvy = m40/m .* [-1.5,0,1.5]; % possible values for vy possvz = m40/m .* [-3.5,-4,-4.5]; % possible values for vz for ndx = 1:3 for ndex = 1:3 for index = 1:3 vx = possvx(ndx); vy = possvy(ndex); vz = possvz(index); v = [vx;vy;vz]; % velocity of the ball [xval, yval, time, height] = solver(r,v,W); XVal(ndx,ndex,index) = xval; YVal(ndx,ndex,index) = yval; Time(ndx,ndex,index) = time; Height(ndx,ndex,index) = height; hold on end end end AvVelocityX = 4 ./ Time; end
function [XVal, YVal, EndTime, SeeTime, Height, AvVelocityX] = ... runge_kutta_drag_eq_solve_w(D)% Solve the diff eq for a ball moving through air %clear %clc %close all global k g K t_step t_max eps color maxVx color = 'r'; % color of the graph g = 9.8065; % gravitational acceleration in m/s^2 rho = 1.225; % density of air in kg/m^3 c_d = .47; % drag coefficient of a sphere eps = .9; % coefficient of restitution for the ball maxVx = 19.21; % maximum speed clearly visible
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m40 = .0027; % mass of 40mm ball Rad40 = .02; % radius of 40mm ball Rad = D/2; % radius of ball m = .337493 * D .^ 1.5; % mass in terms of D % calculated using m and D for 38mm and 40mm ball A = pi*Rad^2; % cross-sectional area of ball in m^2 k = .5 * rho * c_d * A / m; K = 2*pi * rho * Rad^3 / m; v = m40/m .* [20;0;-4]; % initial velocity in m/s r = [-2;0;.7]; % initial positions in m W = [0;0;0]; % initial spin t_step = .001; t_max = .5; XVal = zeros(3,3,3); % x value of bounce YVal = zeros(3,3,3); % y value of bounce EndTime = zeros(3,3,3); % time to reach the other player seeTime = zeros(3,3,3); % time to be able to see the ball Height = zeros(3,3,3); % height at net possW = (m40*Rad40^2)/(m*Rad^2)*[-20,0,20]; % possible values for the components of W for ndx = 1:3 for ndex = 1:3 for index = 1:3 wx = possW(ndx); wy = possW(ndex); wz = possW(index); W = [wx;wy;wz]; % angular velocity of the ball [xval, yval, endtime, seetime, height] = solver(r,v,W); XVal(ndx,ndex,index) = xval; YVal(ndx,ndex,index) = yval; EndTime(ndx,ndex,index) = endtime; seeTime(ndx,ndex,index) = seetime; Height(ndx,ndex,index) = height; %hold on end end end AvVelocityX = 4 ./ EndTime; SeeTime = mean(mean(mean(1 - seeTime ./ EndTime))); % percentage of time % you can actually see the ballend
function [xval, yval, endtime, seetime height] = solver(r,v,W)% Solve the Diff Eq:% v = - g (z direction) - k * v * norm(v) - K * cross(W,v) global t_step t_max eps color maxVx T = 0:t_step:t_max; stepNum = length(T); V = zeros(3,stepNum); V(:,1) = v;
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R = zeros(3,stepNum); R(:,1) = r; go1 = true; go2 = true; go3 = true; go4 = true; % Runge-Kutta method of solving the diff eq for the velocity % Finds the trajectory as you go along for ndx = 1:stepNum-1 t = T(ndx); k1 = t_step .* equation(v, W); k2 = t_step .* equation(v + .5*k1, W); k3 = t_step .* equation(v + .5*k2, W); k4 = t_step .* equation(v + k3, W); r = r + v .* t_step; R(:,ndx+1) = r; v = v + (1/6) * (k1 + 2*k2 + 2*k3 + k4); V(:,ndx+1) = v; vx = v(1); x = r(1); y = r(2); z = r(3); % Add in inelastic, frictionless collisions with the table. if go1 if z < 0 v(3) = - sqrt(eps) * v(3); xval = r(1); yval = r(2); go1 = false; end end % Time to slow beneath maxVx - where you can see it if go2 if vx < maxVx seetime = t; go2 = false; end end % Time to reach the other end of the table. if go3 if x > 2 endtime = t; go3 = false; end end % Height as you pass over the net if go4 if x > 0 height = z; go4 = false; end end
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end % Make plots Vx = V(1,:); Vy = V(2,:); Vz = V(3,:); X = R(1,:); Y = R(2,:); Z = R(3,:); %plot3(Vx,Vy,Vz, color); %grid on %xlabel('Vx'); ylabel('Vy'); zlabel('Vz'); %plot3(X,Y,Z, color); %grid on; axis equal; %xlabel('X'); ylabel('Y'); zlabel('Z'); %plot(T,Vx, color); %grid on; %xlabel('time'); ylabel('Vx'); end function prime = equation(v, W) global k K g prime = -[0;0;g] - k .* norm(v) .* v + K .* cross(W,v);end
