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04052014, 12:23 AM  #21 
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The derivation shows projectile motion with drag in one dimension. The 4.29 ft of fall due to gravity is the amount the ball would have fallen in 0.52 seconds if it had been struck horizontally.
The assumption of a horizontally struck ball underestimates the true serve speed as it does not consider the vertical component of the serve. To estimate the error, we have to consider the angle at which the serve was struck. Consider the angle the racquet makes with the vertical to be theta. Let V be the initial velocity imparted on the ball, then the velocity of the vertical component would be Vsin(theta). This theta varies depending on the height of the player. A taller player with be hitting down the ball more and thus resulting in a larger theta. As an example let's assume theta = 15 degrees. Let W be velocity ignoring the vertical component, then W=Vcos(theta)=0.96V. As we can see, based on the strike angle, the error from the horizontal assumption can be quite small. It would interesting to see the error for the taller serving pros. 
04052014, 07:18 AM  #22  
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Have you ever tested your matlab model against radar? Just thinking off the top of my head, are there any easy corrections that could be made like nootles is attempting above to correct for the time in flight due to being hit down toward the court? I can see that topspin serves are going to be very hard to correct without estimating spin rates. 

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04052014, 10:14 AM  #23 
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After reviewing Maverick's web page, I think both jmnk and I were wrong about the assumptions made for the calculations. The standard practice for projectile motion is to separate the vertical and horizontal dimensions. However, in Maverick's calculation, he assumes the trajectory to be one dimensional. It is assumed that after the ball leaves the racquet, the ball travels in a straight path. This is supported by the way he details an estimate of the distance:
"Estimate of the distance traveled in the air by your serve. The distance from the middle of the baseline to the cross of the service T is 60 feet; the distance from the middle of the baseline to the wide coners of the service box is 61.5 ft. The fact that the serve is hit from a height of 8 or 9 feet increases the distance by 0.5 ft. But typically serves are struck a little inside the baseline, so you have to reduce by that distance. If your serve lands about a foot inside the service line, you have to estimate and subtract that distance as well." As you can see, he is describing how to estimate the distance the ball traveled in a straight path in the air. The added 0.5 ft due to hitting it higher off the ground is his estimate to increase the flight path due to height. With this assumption, the formula actually over estimates your serve speed. The force of gravity, while small compared to drag, still affects the ball. The real trajectory of the ball is not a straight line but a downward curve. If we look at the default calculation shown, the ball traveled 4.29 ft in the vertical direction that cannot be attributed to any factors in the model. Estimation of the error is hard in this case. Because the actual path is a curve, the component of gravity in the direction of the ball is constantly changing. But amount of fall due to gravity can be seen as a gauge for the error. The smaller the fall due to gravity is, the more accurate the serve speed. Maverick decided to neglect gravity because it is a vertical force and thus "doesn't significantly affect the calculation." While it does not significantly affect the calculation, this is actually due to the fact that gravity is a very weak force compared to drag at high velocities. If you softly push a serve, you can clearly see the curved trajectory, and thus the effects of gravity. Aside from errors in the model, I feel we should also address the errors coming from the measurements. The parameters are: frame rate of camera, distance, number of frames, and drag coefficient. The frame rate of the camera should be fairly accurate, so we can assume no error here. The number of frames can only be off by + or  0.5. If the ball lands in between frames 15 or 16, simply choose 15.5. The velocity is inversely proportional to the number of frames. At lower number of frames, the error is greater. The speed calculated using 10 frames will be 1.05 times greater than the speed calculated using 10.5 frames. However, speed calculated at 15.5 frames will be about 1.032 times greater than the speed calculated using 16 frames. For better accuracy, a high frame rate camera is highly desired. At 120 frames, the speed is only about 1.004 times higher than 120.5 frames. I include drag as an source of error because not all environments are the same. I'm not sure as to the error in the measurement of drag. The drag used in the calculation is estimated using a know serve of Sampras, however this only gives the drag in the conditions the serve was hit in. With the amount of studies available, I think we can find a fairly accurate drag coefficient applicable to the most of us. Lastly, the distance traveled. It is important to use the correct distance in the calculation. Because of the exponential term the formula will be quiet sensitive to the distance. If we look at the default calculations on the site, a change of 3 inches, will result in a difference of almost 0.5 mph. 
04052014, 10:48 AM  #24  
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04052014, 11:13 AM  #25 
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The model assumes a linear path, there will be inherent error with that assumption. Working within the model, you should be using a linear path from racquet contact to court.

04052014, 12:46 PM  #26  
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Maverick model assumes that the serve is struck horizontally. And it also neglect vertical component that the ball, struck horizontally, would acquire due to gravity. But indeed that component due to gravity is not going to be that great. However neglecting vertical component due to the serve being struck downward is a bigger error. That is the biggest source of inaccuracy. Quote:
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04052014, 12:48 PM  #27  
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04052014, 01:12 PM  #28  
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The drag coefficient calculated using Sampras' serve was done under these same conditions. 

04052014, 01:28 PM  #29 
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At the very least, if someone uses the frame counting method directly from the web site, he should enter ONLY horizontal distance under "Distance traveled in air".
In other words, if you are assuming that you are hitting the serve from inside 1foot of the baseline, and it hit the court 1 foot before the service line, than you should enter 58 under 'Distance traveled in air'. This is because the formula really calculates horizontal speed component only, so any distance in nonhorizontal direction should not be taken into consideration. As counter intuitive as it may seem, the height the ball was struck from is irrelevant when using this formula. Similarly, when using iphone's servespeedapp, you should enter 0 under 'select player's height'. This should prevent the app from considering nonhorizontal distance the ball has traveled. (Although I do not own this app, so I do not really know what will happen when you select 0 at this step. Or if the app even has option to select 0). 
04052014, 01:35 PM  #30  
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Neglecting gravity is one thing. Assuming horizontal serve is another. These are sort of independent assumptions. the formula does both, and both lead to errors. (@nootles  since I doubt anybody else understands your point, so between two of us. I get your point about changing the frame of reference such that any serve can be considered horizontal (i.e. parallel to the Xaxis.). But if you do that than you need to recalculate the distance from the point of the serve to the bounce point in that new frame of reference. The more the serve is hit downward, the less that distance is going to be in the new frame of reference. the model does not do that. I suppose it could be done, although I doubt you are going to end up with anything simpler. BTW  the best paper on the ball trajectory topic I have found is this: http://goffj.web.lynchburg.edu/Goff_Carre_AJP_2009.pdf. It is on soccer ball, and it has few errors as well, but it is a very solid theoretical discussion of tridimensional trajectory.) You can easily try an experiment. Drop the tennis ball from 9 feet, just a free fall. Enter the distance traveled in air into equation, and enter the time it took for the ball to reach the ground. If the formula was correct you should get 0 as the initial speed. Because at free fall the initial speed in any direction is 0. Last edited by jmnk : 04052014 at 01:46 PM. 

04052014, 01:54 PM  #31  
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The same principle can be applied to Maverick's model. By assuming a one dimensional path from the contact to the landing, the two dimensional problem can be reduced to a one dimensional problem by a rotation of the coordinate axis. In your example of free fall, the error is from neglecting the effects of gravity. Since the model ignores gravity, it rectifies the fact that the particle moved by giving it initial velocity. This is a fundamental error in assumptions that the model is based on, rather than the model itself. 

04052014, 02:04 PM  #32  
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" @nootles  since I doubt anybody else understands your point, so between two of us. I get your point about changing the frame of reference such that any serve can be considered horizontal (i.e. parallel to the Xaxis.). But if you do that than you need to recalculate the distance from the point of the serve to the bounce point in that new frame of reference. The more the serve is hit downward, the less that distance is going to be in the new frame of reference. the model does not do that. I suppose it could be done, although I doubt you are going to end up with anything simpler. " 

04052014, 02:27 PM  #33  
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Suppose you are standing at the center mark and serves down the tee and lands right at the corner. Suppose to make contact directly above the center mark 8 ft off the ground. The horizontal distance from center to tee is 60 ft. By the Pythagorean Theorem, the distance between the two points is sqrt(60^2+8^2), this is approximately 60.53ft. So Maverick indeed does tell you to calculate the distance of the flight path and not just the horizontal distance traveled. 

04052014, 03:25 PM  #34  
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On the other thread, there were many estimates using the calculator, but it wasn't at all clear what distances were being estimated and entered. I believe with one video I calculated it 5 mph slower than someone else (based on the given frame count), most likely because I estimated a reduce distance. 

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04052014, 03:33 PM  #35  
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