Long Drive over Niagara Falls

Executive summary

• I have no doubt that a Maurice Allen's good competition drives could have cleared Niagara Falls based on carry distance.
• There are certainly causes for skepticism that his drive actually did carry from Canada to the USA as shown in the video.
• In the end, I feel it is more likely than not that the drive did cross the Falls. But I don't believe it was at all like the video production depicts. I am convinced what the video iimplies is the ball landing on Terrapin Point is actually the second bounce, not the initial landing.
  

Setting the stage

Here is an aerial view of the falls, showing the Horseshoe (Canadian) Falls and the American Falls. Maurice Allen's drive (the red arrow) uses the narrowest crossing between Canada and the United States in the vicinity of the waterfalls. Allen is hitting from a platform on the Canadian side near the tourist center, and aiming at the Terrapin Point observation area on Goat Island right next to Horseshoe Falls.

On the left is a screenshot of the launch area in Canada, showing the blue-carpeted platform where Allen will drive from. On the right is the landing area across the Niagara River on the American side, the observation park at Terrapin Point. Note that both are on really tall cliffs; we will see consequences of that terrain as we go through the exercise.

Measuring the site

The first and most important dimension is the distance from the launch area to the landing area; that is the minimum carry distance the drive is allowed to be. Any shorter a carry, and the drive hits the face of the cliff and bounces back into the Niagara River. Imagine a Pete Dye course with rock cliffs instead of wooden railroad ties.

Both Google Maps and Google Earth have the ability to measure straight-line horizontal distance; here is the Google Maps version, and the distance is 341yd (that's 1023ft divided by 3). That is almost the same as Maurice Allen's own laser rangefinder measurement of 342yd.

The "official" distance of the drive was 427yd total distance. That's the stated distance on the plaque, and the one Allen cites when asked. But my assignment was more like, "Is is possible for Maurice Allen to drive across Niagara Falls?" That means carry distance, not total. For purposes of this analysis, let's use 350yd as our target carry distance; Allen has to be standing severall yards behind the cliff to hit the ball, and the ball has to clear the railing and the ball has to land on solid ground.

Let's mention here that there is a carry distance reported in several accounts of the feat. The distance is 393 yards. That is the distance taken at the spot the ball appeared to land in the video. It is not cited in the video itself nor inscribed on the plaque, but we can find it in reports in Golfweek (USA Today), Wikipedia, and some news reports. As far as the public is concerned, that seems to be the "official" carry distance.

I also looked at altitude, for two reasons:

• The air density will affect the aerodynamic lift and drag on the ball, and therefore the distance.
• Any difference in height between launch and landing will bring different requirements to the launch conditions.

Google Earth allows you to query the elevation of a point you select. But the numbers for elevation on both sides of the Falls were very unstable; small movements of the point result in large changes in the indicated elevation -- much more than the video of the terrain would suggest.

Based on this data, let us assume that the elevation of both launch and landing are at about 400ft. We will use altitude for aerodynamics, but not consider any elevation difference when we compute the trajectory. (Frankly, I think there would have been a mention in the video of an elevation difference, if indeed there is one. That would be a significant aspect of the challenge.)

Measuring Maurice Allen's norms

• A number in red means it was computed from other, given, data. If it is in purple, the only computation was unit conversion.
  
• The numbers available are total spin. We will use it for backspin. The maximum likely error is about a half percent (around 10rpm), which is negligible for our purposes. The error was computed assuming a maximum spin tilt of 6° and spin of 2000rpm..
• Some sources had total distance instead of carry. That is of no interest for this problem; it doesn't matter how far it runs out after it lands, just that it lands far enough to carry the Niagara River.
• I computed smash factor in order to make sure we could use an ideal impact model. Four out of five were right on the theoretical maximum, and the other was not too far off. So yes, we can use the impact model for Maurice's swing, without making allowances for imperfect strikes.
  

The video itself

• "Highlighting golf's fun, human, and humorous side."
• "...don’t miss out on the most entertaining golf content online."

My analysis tool

• An impact model is a mathematical process that takes impact parameters like clubhead speed and loft, and converts them into launch parameters like ball speed, launch angle, and spin. BTW, TrajectoWare Drive is the only program I know that has an impact model that can also be applied backwards. That is, like many trajectory programs, you can set the impact parameters and see the resulting launch parameters. (Every impact model can do that within its own constraints.) But with TrajectoWare Drive, you can set launch parameters and see what the impact parameters must have been in order to caust that launch.
  
• A ball flight model is a mathematical process that takes launch parameters, along with environmental parameters like wind speed and direction, and converts them into a path that the ball follows in its flight. The path is called a trajectory. The most important output for our purposes is the carry distance, but it also gives other information like maximum height, hang time, and angle of descent.

Launch parameters

1. In the competitive examples, his ball speed is in the 210-216mph range, with one drive at 221mph .
2. In his practice the week of the Niagara drive, his ball speed is in the 193-203mph range.

1. Of course, the carry distance must be at least 350yd.
2. The max height of the trajectory must be between 150ft and 196ft.
   
3. We have to have all the other parameters inside Maurice's normal range:
• Ball speed was discussed above: runs at 214mph and 203mph.
• Launch angle 12.6-14.1 degrees.
• Spin between 1550 and 2200rpm.
4. The environmental variables are:
• Temperature = 70°F. (No information about that in the video, so I'm guessing.)
• Altitude = 500ft above sea level. This assumes hitting at 400ft as discussed earlier, and the ball spending most of its trajectory considerably higher. We are assuming here that its average elevation above the ground will be 2/3 of the maximum height. (That is the geometry of a parabola. The trajectory is not a parabola, but its average statistics should be similar.)
5. I am not figuring in:
• The mist. (I don't know how to model mist. Later we will make a worst-case assumption about the mist's effect to apply some skepticism to the claim.)
  
• The fade that Maurice said he is trying to play.
• The wind. (We don't know enough about it, and it was not flagged as much of a factor in the video.)
  
If it turns out to be important, TrajectoWare Drive can model the fade and the wind. It can't model the mist.

• Carry distance = 366yd. That is more than enough to carry the Niagara, with a margin of 18yd over our 350yd target, which itself has a 5-7yd margin built in.
• Max height = 55.1yd = 165ft. Within range for a Maurice drive.

• Launch variables: ball speed = 203, launch angle = 14.1, backspin = 2200.
• Carry distance = 351. It is barely over our target of 250. But it is over, and don't forget that 20yd bonus that we haven't factored in.
  
• Max height = 59.6yd = 179ft. Within range for a Maurice drive.

Impact parameters

First cut

• Clubhead speed = 144mph will require one of Maurice Allen's best swings; it's possible, but not a cinch by any means. OTOH, 137mph should be very easily achieved by Maurice; way below his good competition swings.
• Angle of attack (AoA) = both around 9° This sounds quite high. We'll look at it next.
• Spin loft = 4.0° is the loft of Allen's Krank driver. Factor in shaft bend and the spin loft will be a little more.. We're in the right ball park here. (Yes, this is called "dynamic loft" on the TrajectoWare screen; see the footnote for explanation.)
  

Reality check: AoA

• Increasing launch angle by increasing loft increases spin.
• Launch angle from loft is only about 85% efficient in converting loft to launch angle. (That is, 10° of loft only produces 8.5° of launch angle. The rest goes into spin.)
• Every degree of AoA goes into launch angle, and adding AoA does not add any spin at all.

• In a Golf Magazine article from 2019 (the same year as the Niagara feat), Luke Kerr-Dineen shows TrackMan numbers for Tim Burke, another elite Long Drive champion. The numbers include an attack angle of 7.9°. That is high enough to believe that 9° is indeed humanly possible. The biggest difference between Burke's numbers and our launch condition for Niagara was a spin of 1850rpm instead of 1550; the speeds and launch angles were pretty close.
• I was able to estimate Allen's AoA from another video of his swing. It came out at just under 6°, much lower than I had hoped. 6° is still a very high AoA for tournament golf if not Long Drive, but it is not close to the 9° needed to produce our launch parameters.
  

• Clubhead speed = 140mph. Allen can certainly do this. It is better than his practice earlier that week, but still below his good drives in competition -- and well below his winning drives.
• AoA = 8°. This is a good number for a Long Drive, and Allen's record certainly says he is competitive.
• Spin loft = 5°. That is 4° of loft built into his driver and and extra degree due to shaft bend.
  

• I don't see such a drive carrying 393yd as claimed. That would take one of his best drives to accomplish.
  
• There are other factors that invite skepticism. Let's look at those next.
  

Ah, but did it really?

• The effect of the mist on carry distance is probably adverse, and we have no estimate of how much.
• The elapsed time ("hang time") in the video is longer than it would be for any of the modeled drives. It is also longer than experience at Long Drive competitions supports.
  
• The angle of descent appears much too steep.
• Why does the video show no launch or impact data for the purported successful drive? Indeed, shouldn't this "stunt" have been well observed and instrumented to prove it happened?
• How lucky did the Skratch team have to be to have a camera in the right place and pointed in the right direction to have caught the ball as it landed?
  

Effect of the mist

• A piece from Golf Magazine (2020) consisting of anecdotes from PGA tour players.
• Tour Spec Golf (2009) tested a variety of clubs and balls, to compare wet vs dry performance.
• Today's Golfer (2021) tested to compare wet vs dry balls when hit with PW and Driver.
  

Hang time

• Elapsed time on the video - Using the downloaded video, I located two key frames:
  
  1. The instant of impact (launch) in the frame at 9:28:23.
  2. The orange streak of a ball lands at frame 9:39:24.
     
  The time difference between them (simple time arithmetic at 30 frames per second) is 11 seconds plus one frame, or 11.03sec.
  
• Hang time of modeled drives - All of the modeled cases shown above have hang times around 8sec. (I did a lot more runs than just the ones shown here, and all their hang times were also between 7.7 and 8.54sec.)
• Experience with Long Drive competition - Experienced long drivers also called out this discrepancy. They did not base it on a mathematical model, but rather years of experience in Long Drive. And that experience says that a hang time much over 8sec is very rare. In support of that opinion, Allen's winning drive in the 2018 World Championship hung less than 9sec. (I measured it as 8.8sec, and it was very high for a Maurice Allen drive.)
  

Angle of descent

Here is the frame at 9:39:22, two frames before the ball bounces on a paved walk on Terrapin Point. If you click on it, you will see an enlarged copy of the landing area alone. I chose this frame because it had the clearest trace of the ball (the orange streak just right of center). The single frame before impact was also clear enough to be measured; the streak was closer to the ground and not as vivid.

I measured the angle of descent (AoD) in both frames. It was just over 71°.

This is a problem! 71° is extremely steep for an angle of descent. All the modeled trajectories had an AoD between 43° and 47°. An AoD above 70° just is not going to happen.

But wait! There is certainly a way to reconcile the 71° that we see with the 45° that it should be. It turns out we need to investigate where the camera is and where it is pointing.

We actually don't know that the camera is at right angles to the path of the ball. It certainly looks that way to our eyes. I too assumed that it was perpendicular the first time I looked at it. I'll get back to that shortly, but simply assuming that was a mistake on my part. Natural perhaps, but wrong.

And it matters. If the camera angle is more parallel to the path, then it will appear to be a higher angle of descent. Look at the diagram, which shows the descent of the ball (in orange) to the ground (gray). The camera pointing perpendicular to the path will measure the actual AoD, say 45° or whatever the AoD really is. But the camera aimed right up the path and parallel to it will always measure the AoD as 90°. Camera angles in between will give angular measurements in between.

For instance, a camera angle 21° from parallel will turn an actual AoD of 45° into what looks like an AoD of 70° -- which might possibly be the situation we have here. (A derivation of how to compute this is in the footnotes.)

Why should we worry about this? It sure looks like the camera is aimed across the path of the ball. As I said earlier, I made that mistake at first, too. But look at the video from 9:39 until the ball lands in the grass 5 seconds later. The camera pans dizzylingly fast. It has the look, to me at least, of an extreme telephoto panning. And extreme telephotos (a) are misleading to the eye about where they are located and (b) greatly foreshorten anything with depth to it. So it could easily distort the picture to look like it is square-on perpendicular when it is not.

You don't have to take my word about that; I'm just telling you what gave me the hint to look further. But when I looked further, things began to fall into place in a pretty unambiguous way. It is possible to find out what the camera angle really was. The thing we need to do is figure out where the ball landed and where the camera was. Sounds difficult, but let's try.

Where did the ball land?

At that point, I drew a few lines on a Google Earth screenshot. (You can click on it and see it fullsize.) The red and blue lines in this aerial or satellite photo are drawn through those details covered by the red and blue lines in the frames above. The lines meet where the camera had to have been to show both views. The process is called "triangulation", in case you ever heard that term used.

Once I had triangulated the camera's position, looking at a few more frames made it very clear where the ball bounced (the orange circle). As further confirmation, the orange dot would be only a little less than 400yd carry -- close enough to the claimed 393yd. The claim is clearly based on where the camera saw the ball bounce

The orange broken line is my best guess as to the path of the ball. It is a straight line from the platform on the Canadian side to the place where the ball bounced.

It is clear just from looking that the camera angle is far from perpendicular to the path; in fact, it is much closer to parallel. The angle between them, according my trusty protractor, is 22°. Suppose the actual angle of descent is 45°, consistent with what we would expect. When I go throught the math, I find the apparent angle of descent (as seen by the camera) to be 69°. That is not far from the 71° we were worrying about. Given the error sources in this whole process, I no longer feel that the AoD is reason to be skeptical about the drive. I know the camera is looking much closer to down the path than perpendicular to it -- no matter how it appears in the video.

But we still have a problem, and a big one!

We now have a really good idea where the ball came down. And that is almost 400yd from the launch platform! It is not hard to believe a carry distance of 350yd. But a 393yd carry is a lot to ask; it is a championship-winning drive. Allen may be capable of it, but not often. Lots of room for skepticism here.

Does that mean it was faked? Maybe.
But there is another possibility.

The second bounce theory

• The too-long carry distance. The ball got to 393 yards not on carry alone, but by a big bounce off a paved surface after a significantly shorter carry.
  
• The too-long hang time. If the first bounce was enough earlier, the hang time could be exactly what we would expect.

• Once I knew exactly where and when to look for the ball, I found a faint orange trace that could only be the golf ball, landing and bouncing again. I could only see it stepping through a frame at a time; it was that faint. But I was able to associate the visible bounce with a single frame
  
• When I listened closely, dragging the cursor a frame at a time, there was a faint but definite click close to the time the ball landed. Why dragging the cursor? Because the single-step button on my editor does not play the sound track.
  
• I listened for a click about 3sec earlier, again dragging the cursor slowly through from about 2.7sec to 3.3sec before the bounce. I was able to find a faint but audible click. Why was I looking for a click 3 seconds earlier? Because we needed evidence of a first bounce that wasn't the one we could see. OK, but why 3 seconds? There is a 3sec discrepancy between the 8sec hag time of a normal long drive and the 11sec elapsed time in the video. 3sec earlier would erase the hang time discrepancy.

But were the clicks really there? Both of them? Or was that just my own imagination attributing them to random stuff on the video sound track. I felt I had to be sure the clicks I heard were not random artifacts that I was wishfully interpreting as a ball bouncing. So I looked at an oscilloscope trace of the sound. That was easy to obtain, because my video editor has a built-in sound editor that includes an o'scope display.

In this screenshot, note that the sound power is dominated by a steady noise, the roar of the Falls. But there are two very narrow sharp spikes (pointed to by yellow arrows) at exactly the moments when I heard the clicks. Their peak values are the highest of any amplitude on the display. They are surely distinct clicks, not random phenomena nor the product of my imagination, so I have some confidence both are bounce sounds of a golf ball on pavement.

Yes, after the second click, there are other events visibly above the ambient noise from the Falls. But listening to it you can tell they correlate to human voices and seagull calls.

I looked very carefully at the placement of events, a frame at a time. There was a slight anomaly that bothered me. The second click on the sound track was a little more than 3 frames later than we see the bounce in the video. Why should we have that lack of synchronization? More processing in the camera for sound than video? (Not likely; video should take more processing than audio.) Sloppy "manufacturing" of a fake video? (Possibly, but that would seem to be something easy to get right.) Much lower speed of sound than light? Let's look at that further.

To check out an assertion about the speed of sound, we need to know distances from the bounces to the outtake camera. I found the camera's position by triangulating again, this time from the outtake video. Here is a map showing the outtake camera (this time in bright green), my guess at the first bounce way up at the tip of Terrapin Point, and the second bounce which we have already located pretty well.

The footnotes have a table of distances among the three interesting places in the picture. From the second bounce to the camera that recorded it is 126ft. It takes sound 115 milliseconds to go 126 feet (also in the table). The video was shot at 30fps, so a frame is 33msec long. That means the speed of sound alone accounts for 3½ frames -- which is exactly the size of the anomaly we are trying to explain.

Mystery solved!

And, at this point, I'm beginning to have a lot of confidence in the outtake as evidence for the two-bounce theory.

• The ball carried just under 350yd. That is almost 400yd to the second bounce, minus about 53yd representing the distance the ball bounced after it first landed.
• The hang time was 8.06sec. That is 11.03sec from launch to second bounce, minus 2.83sec between first and second clicks, minus another 0.14sec to account for the speed of sound.
• The angle of descent coming down after the first bounce was around 46°. Remarkably consistent, considering (a) the uncertainties in the exact camera angle, and (b) this was coming off a bounce on pavement, not on the fly.
  

Attention to detail

Luck or fake?

1. A lot of cameras to cover much of Terrapin Point. (Expensive, and I saw no evidence of that.)
   
2. The ability to track the ball with one or just a few cameras, so the camera would be pointing at where the ball lands when it lands. (Unless it was radar tracking, the mist would prevent tracking it until it got pretty close to Terrapin Point.)
3. A lot of luck. (Can't rule that out, but it's not the way to run a "one chance is all you get" like this.)
4. One or a few cameras with a wider angle and higher resolution, then use "digital zoom". (Might change the need for luck to needing just achievable skill.)
   
5. Someone or something out of sight in the landing zone to toss a ball where you have a camera, then claim that's the ball Maurice hit. (That is what some skeptics think. What would I need to find out to convince myself that didn't happen?)
   

1. Lots of cameras - Skratch might conceivably have done this, but I doubt it. Still... At least one of their cameras was a GoPro, so they might conceivably have sprinkled the grounds with GoPro cameras and used the clip from the GoPro that happened to catch it. The reason I doubt this is that the camera that caught the landing was moving to follow the ball. That rules out a stationary GoPro, at least for that camera. But that brings us to...
2. Tracking - The camera whose footage was eventually used started moving before the ball had bounced, and continued turning and trying to follow the ball until it settled in the grass. That is not a fixed GoPro, but a camera either manned or with a very sophisticated tracking system. Ball-tracking cameras do exist; major-channel golf coverage uses them. If they use radar tracking (like TrackMan technology), the mist probably would not bother them too badly. Would the camera used by Skratch that caught the ball landing have that technology? I don't know. I would assume no, but that is mostly guess. Here are a couple of hints, some suggesting it was tracking and others suggesting it was not.
• Even in the mist, you can make out the banners on the Canadian side. And they disappear downwards in good sync with the moment Allen hits the ball.
• The camera's aim is hard to tell, given the mist. But about a second before the ball lands it is panning down and left, past the streetlights.
• It gets to the landing spot before the ball, then the ball falls into the frame; I don't think tracking cameras work this way. At least the ones I have seen on TV do not.
• It still points to the sky at 8 seconds, when the first bounce would have occurred; we don't se anything until almost 10 seconds. (We see the streetlights first.) That is inconsistent with the second bounce scenario, and I don't believe the first bounce narrative because of both timing (makes no sense) and distance (possible, but much less likely).
  
3. Losta' luck - The chance that a single tight-telephoto camera caught the ball bouncing is pretty small. Not zero. Not completely prohibitive. But small. I would not expect a professional golf video company like Skratch to have taken this sort of chance. I kind of doubt that it was just luck. If that is the only explanation (e.g.- no tracking and no digital zoom), I have much less belief that the ball actually crossed.
   
4. Digital zoom - (See the footnote for how digital zoom works. This is also very likely the way the zoom in your phone camera works.) It uses a video camera with lots of resolution and a normal to wide-angle lens. Then the picture is cropped down to the part of the frame that contains interesting stuff -- like the ball landing. I could believe this without too much trouble.
   
   • When I look at the right half-frame from the camera that took the ball landing, I see an unsharp image. That might be because it is a small fraction of an HD or even 4K frame. That is because digital zoom is less zooming than cropping; it simply reduces the resolution.
     
   • It is fairly easy to believe that the videographer heard the first bounce, and was able to find and manually track the ball in 2 seconds so that the last second captured the second bounce. The field of view could be a very manageable "normal" (e.g.- 50mm) lens -- then digitally zoomed to the image we see in the video.
   • If the digital zoom was a 4x zoom, it would look like a pretty tight telephoto (200mm lens), similar to what the Skratch video appears to be.
     
5. Fakery - And if not just luck, did Skratch determine the outcome in advance and make sure that is the way it happened? Was there someone out of sight of the camera or outside the frame, who tossed a ball with a high arc into the area where the main camera was pointed? We can't rule this out. Without a fairly strong case for one of the first four possibilities, it seems the most likely. With that in mind, I looked hard at the video and also the outtake for such a culprit. I did not find one. But here are things I could not check.
• In the video itself, there are pictures where they showed Terrapin Point mixed in with the drive footage. I was looking for a human with the main camera where we know it was, and also for a human downhill who might have thrown a ball in the air. I never found either one. If I couldn't  even find the videographer, my most likely conclusion is that those clips were not made during the time Allen was hitting drives. They are probably "stock shots" that Skratch took earlier or later.
• In the outtake, there was a bluff that hid the ground below where the ball landed. Could someone have been hiding in that hollow and thrown out a ball? Perhaps, but unlikely. It would have been difficult to get enough speed and direction on a ball from the crouch that would have been necessary to stay out of sight.
Does any of this mean that the video was faked? Not at all. But it is a case for not ruling out deception.

The verdict

• The presence of two clear and objectively "there" clicks, which support the two-bounce hypothesis. and the two-bounce hypothesis explains the hang time discrepancy.
• Two bounces discredits the official narrative. I doubt Skratch would deliberately discredit that part of the story in a fake outtake video.
  
• If I were trying to fake the clicks, I would not have made them so hard to find -- lending credence to their being genuine.
  
• Little touches like allowing for the speed of sound. It took me a while to figure that one out. If the outtake was faked, they went to a lot of trouble to drag along a fact checker, and assumed the fact checker would know a lot of physics.
  

Conclusion

• The elapsed time ("hang time") in the video is longer than it would be for any of the modeled drives, and longer than actual experience in Long Drive.
• The angle of descent appears much too steep.
• We don't know how the mist might have affected the ball flight. I plugged in a 20-yard penalty based on very questionable anecdotal evidence; the actual penalty is likely less than 20yd.
  
• Where did the ball actually land? That detail could validate or cast serious doubt on the drive. The general belief that the ball's first bounce was at 393yd where the camera caught it is not credible -- in the realm of possibility, but not likely.
  
• There was no instrumentation present to document the parameters of the drive. With a "stunt" of this magnitude, I would imagine they would want the technology right there.
• The narrow-field telephoto lens that caught the ball landing had to be incredibly lucky to be in the right place and pointed in exactly the right direction when the ball landed. I'm not sure I buy that, as opposed to some faking going on. In the end, this is my most skepticism-worthy objection.
  

Footnotes

1. Unstable elevation - Here is my guess at the reason for the unstable elevation readings. Google Earth probably has a coarse grid of elevation numbers every few tens of yards or so, and any point's elevation is interpolated between those known points. But both the launch and landing areas are on the edges of cliffs, so there is a huge difference between the elevation on one side of the cliff and the other. (The falls are over 150ft tall, which is close to the difference in elevation between the top and bottom of the cliff on either side of the falls.) That would result in small movement on the ground resulting in large differences in the interpolated elevation.
2. Altitude assumptions - I realize I am being somewhat cavalier by pegging an assumed, round-number altitude. There are several reasons:
1. There is so much conflicting data. For instance:
• The official listed elevation of Niagara Falls is 325ft. I don't know where on or in Niagara Falls this is. Note that the falls themselves are about 150ft top to bottom.
  
• The unstable elevation readings from Google Earth includes elevations of 325ft to over 500ft between points only 65ft apart -- and that is on the flat paved area around the launching platform! (See 10:34 on the video for the area I'm talking about.)
2. I could not find an online topo map with the resolution we would need to answer the question.
   
3. A few points in the video mention the peak height of Allen's drives; they range from 149ft to 196ft above the launch. So the ball goes through altitude changes in flight which are larger than reasonable differences being argued for the base elevation.
4. It does not make a lot of difference. For some typical long drive launch conditions, the difference in distance between 300ft and 500ft of altitude is only 1.3yd. (That's 360.7yd vs 359.4yd.)
3. TrajectoWare Drive - The TrajectoWare Drive program was developed in 2007 by Frank Schmidberger and me. You can download your own copy from the web site.
   
   As noted in the text, golf balls carry much farther today. Below is a table of several runs of TrackMan's ball flight model. TrackMan has the resources to redefine the model every year, and reams of data to validate the annual new model. So we can expect it to be more accurate and more up-to-date. Here is how TrajectoWare Drive compared with TrackMan, for the three sets of conditions I have been shown.
   

   		Case 1		Case 2		Case 3
   Launch	Ball speed	217		217		214
   	Launch angle	11.7		11.6		13.0
   	Spin	1465		1464		1235
   Environ	Wind	0		8.7mph across		8.7mph across
   	Altitude	0		324ft		324ft
   	Temp	75°F		70°F		70°F
   Results		Track Man	Trajecto Ware	Track Man	Trajecto Ware	Track Man	Trajecto Ware
   	Carry	393	364	393	364	393	360
   	Side displace	0	0	54ft	54ft	52ft	52ft

   Note the difference of just about 30yd of carry, with TrajectoWare Drive giving shorter distances than the TrackMan model. The effect of crosswind is essentially the same. The impact models were not exactly the same, but really pretty similar. So my conclusion is that the main difference is the ball aerodynamics have improved a lot since 2004, when TrajectoWare's ball flight model was conceived.
   
   
4. Spin loft - By common usage today, "spin loft" refers to the loft relative to the vertical clubhead path (the "angle of attack" or AoA). That nomenclature was put in place by TrackMan, which has become pervasive in the golf instruction community. But when TrajectoWare Drive was developed in 2007, TrackMan was not out there in large numbers. It was far more a research tool than used in general instruction. Frank and I did not know the TrackMan nomenclature, and we needed names for the quantities we were dealing with. Unfortunately, "dynamic loft" means something else in TrackMan terminology, which is confusing. Sorry 'bout that! If and when there is a need for a new version of TrajectoWare Drive, it will fix the differences to correspond to TrackMan conventions. Until then you'll just have to mentally note that TrajectoWare's "dynamic loft" really refers to "spin loft".
5. Estimating AoA - We could estimate AoA by capturing frames of a video coming into impact, and comparing the position of the clubhead at impact with the frame before impact. But it was hard to find a face-on video of Maurice's swing (most available are down-the-line shots, and that doesn't tell us AoA) sufficiently slow motion to get a position of the clubhead; I found only one where the clubhead wasn't just a blur. I was able to capture three frames: impact, the frame before impact, and the frame after impact. By comparing the position of the clubhead in two successive frames, you can compute the vertical and horizontal differences frame-to-frame. The vertical difference divided by the horizontal difference is the tangent of the AoA. Using the approach (with differences in pixels, measured in an image editor that offers <x,y> values for the cursor), I got:
• Before impact:  11px/108px = tan( 5.8° )
• After impact: 9px/87px = tan( 5.9° )
Given the granularity of the data, we have roughly a 6° angle of attack.

6. Apparent AoD - Let's derive the apparent Angle of Descent when the camera angle is not 90°. (Not that I expect many to be interested. But I do want to be able to reproduce the calculations if I need to.)
   
   
   The diagram shows the descent path of the ball (in orange) and a camera observing at an angle a to the path. We can take any section of the path where it is approximately straight, of height h and base x.
   
   The camera sees the orange triangle with the same height, but a foreshortened base. The apparent base y is perpendicular to the camera sight line.
   
   So the actual AoD, D, is given by:
   tan D = h / x
   
   The apparent AoD, d, is the same, but with the apparent base as seen by the camera. So it is:
   tan d = h / y
   
   Finally, trignonometry tells us that:
   sin a = y / x
   
   So fairly simple algebra gives us:
   sin a =  tan D / tan d
   
   From this we can find the camera angle for a given actual and apparent AoD, or the apparent AoD for a given actual AoD and a camera angle. For instance, our current problem is D=45° and d=70°. So:
   tan D / tan d = 1.0/2.75 = .364 = sin 21°
   The camera angle would have to be 21° to create the foreshortening we see.
   
7. Events in the outtake clip - I wanted to record the exact time (in frames), in case I need to go looking for them again.
   

   Event	Type	Time (frames)
   1st click	Audio	02:29
   2nd click	Audio	05:24
   Ball bounces	Video	05:21

   
   And another table of the distances estimated from plotting the outtake events on a map. Also the time for sound to travel each of those distances.
   

   From position of...	To position of...	Feet	Yards	Time for sound to travel
   1st bounce	2nd bounce	158	53	144ms
   1st bounce	Camera	285	95	259ms
   2nd bounce	Camera	126	42	115ms

   
   While we're here, let's note that the speed of light is 874,000 times the speed of sound. So we can ignore the time it takes for light to travel; for this problem, it is instantaneous.
8. DIgital Zoom is a technique used widely in cell phones to give the effect of zoom without zoom lenses. (Zoom lenses are the system known as "optical zoom".) Digital zoom consists of taking the picture with a wider angle than your photographic intent calls for, then cropping down to the part of the frame you really want.
   
   
   
   Here is a simplified example. The "original" image is taken through a window screen; think of each square you see through the screen as a pixel, an element in the camera's sensor. The "zoomed" image is cropped from the original, then blown up to full size. We don't have any more information about those pixels we just blew up, only fewer pixels than the original. So the picture has lost resolution even though part of it gained size. This only works well if the original had so much resolution that it could afford to lose quite a bit and still look OK.
   
   How about an example with numbers. We have a 4K video frame (for most 4K implementations, that's 2160p in the traditional HD terminology). Suppose we cropped out a 540p chunk of the image, then blew it back up to 2160p. Here are some of the implications:
   
   • That is a 4x zoom; 2160=4*540.
   • If the original was taken with a 50mm lens (a "normal" lens), then the zoomed image will look like it was taken with a 200mm lens (that's quite a long telephoto), except much lower resolution than if it had been done optically -- that is, with an actual telephoto lens..
   • How much lower resolution? Each pixel in the original image is the size of 16 pixels in the zoomed image. That's a magnification of 4x squared, because each zoomed pixel is a 4x4 super-pixel made from one pixel in the original.
   • The result might look pixellated or it might be smoothed by digital filtering, either in the camera or post-processing. Most digital zooms do some filtering. Filtering will usually cause blurring, though some algorithms can identify edges and keep things reasonably sharp.