26 December 2013

Great Christmas Day Dunks from LeBron James!

The NBA does a great job putting games on Christmas Day that showcase the league's biggest stars.  One of yesterday's games had the Los Angeles Lakers hosting the two-time defending champs, the Miami Heat.  LeBron James had a few fantastic dunks during the game.  My favorite were two he got in the first half, both of which were assisted by Dwyane Wade.

With just under four minutes left in the first quarter, Wade drove into the middle of the key and threw the ball up after he was only a couple of feet into the lane.  James was flying in from the left and caught the ball when it was more than five feet from the basket.  The image below shows James just as he caught the ball from Wade (click on the image for a larger view).
James left the court at nearly 10 mph and took about half a second to leave the court, grab the ball, and slam it into to the basket.  He threw the ball into the hole at a speed of about 36 mph.  The ball was rotating at close to 145 rpm with respect to his right shoulder.  That rotational speed is about one third of the rotational speed of helicopter blades.

Moving on to under three minutes left in the first half, James had a dunk that was even more impressive that the one I showed above.  Wade drove down the right side of the key and after going about four feet passed the foul line, tossed the ball up with his right hand.  James was driving into the lane, just slightly left of center.  Wade's toss actually went off the backboard before James caught it, and then followed with a dunk that had his head under the basket as the ball went through the hoop.  The image below shows James just as he is catching the ball from its flight off the backboard (click on the image for a larger view).
As with the first dunk I described, James left the court at nearly 10 mph and was airborne for about half a second before throwing the ball into the hoop.  The ball was moving about 13 mph away from the hoop when he caught it at the instant shown above.  James then reversed the ball's trajectory by accelerating it to about seven times the magnitude of the acceleration due to gravity, or, simply, 7 g's.  He threw ball into the basket at a speed of nearly 23 mph.

Both dunks described here require an extraordinary amount of athleticism.  There may only be a tiny number of people walking the planet who could do what James did.  Those dunks were certainly Christmas treats for those of us watching!

05 December 2013

The Brazuca ball is here!

A couple of days ago, Adidas unveiled the ball that will be used in the 2014 World Cup.  Fans voted last year to name the ball Brazuca.  Check out the colorful ball below (click on the image for a larger view).  The image is a cropped version of an image found at Footy-Boots.
For closer views of the ball, check out more images over at Footy-Boots.

As a sports physicist, I am intrigued by the six thermally-bonded textured panels.  Recall that the Jabulani ball used in South Africa for the 2010 World Cup had eight thermally-bonded textured panels.  The 2006 World Cup in Germany employed the Teamgeist ball, which was the first World Cup ball that did not have 32 panels.  The Teamgeist ball had 14 thermally-bonded panels, but those panels were not textured.

As panel number decreases, textures have to be added if the ball is to have aerodynamic properties similar to balls with more panels.  Making balls smoother with fewer panels leads, perhaps counterintuitively, to larger air drag.  Adding textures to the panels roughens the ball's surface and reduces air drag.  The idea is to add enough texturing to the panels to compensate for the fewer seams when panel number is reduced.  No two World Cup balls have the same aerodynamics properties.  It will be interesting to hear what players have to say about the ball after early World Cup matches.  There always seems to be some whingeing.  Just look back at the start of the 2010 World Cup!

There are 189 days before the 2014 World Cup begins.  I can't wait!

15 November 2013

Fun Time at University of Vermont

My first trip to Vermont was made possible by an invitation from Adrian Del Maestro to give the physics department colloquium on Wednesday, 13 November 2013.  I got to know Adrian early in the current semester when he contacted me about a physics of sports course he is now teaching at the University of Vermont.  I was flattered to learn that Adrian uses my book in his course.  During my brief visit to the University of Vermont, Adrian introduced me to some of the neat things he is doing in his sports physics course.  I also learned about his fascinating condensed matter theory research into systems with strong interactions.

The physics department videotaped my colloquium talk.  I spoke about the Tour de France research I've been doing over the past two summers with Lynchburg College physics major Brian Ramsey.  I also spoke about work on soccer-ball aerodynamics I've performed with Matt Carré at the University of Sheffield.  A YouTube video of my talk appears below.
I need to visit Burlington again!  The snow on the Green Mountains to the east made me want to put on some skis.  The view to the west with Lake Champlain and the Adirondack Mountains in New York state was impressive.  There is also a great selection of beer from local microbreweries.

31 October 2013

Physics Behind a Softball Pitch

If you already miss watching baseball less than a full day since the Red Sox won the World Series, head out to the softball field for a little fun and exercise.  Beware, though, if you get into a fast-pitch softball game.  Swing quickly because the ball will be on you in a hurry!

To examine a softball pitch with a physicist's eye, I enlisted the help of my experimental colleague Will Roach.  We set up a camera to film Lynchburg College softball pitcher Hope Johnson during her warm-up in the bullpen.  The animated clip below shows Hope firing a two-seam fastball.
Blogger won't let me upload the animated GIF, so I went for the movie format instead.  I've identified two tracks in the video.  The one in yellow shows the path of the softball.  The one in red shows the path of Hope's shoulder.  The reason for the second path is to use the shoulder as a reference frame for rotations about the shoulder.

There is so much great physics in that video!  Hope begins with the ball in her mitt.  She then rocks back slightly as she pulls the ball out of her glove.  Note early in the video that her back (left) shoe has the toes off the dirt.  The soft rocking generates a little momentum that she uses as she initiates the full motion of the pitch.

A little later in the clip, Hope is moving forward at about 3 m/s (11 kph or 6.7 mph).  Note her front (right) foot pushes off the pitcher's rubber.  That added push helps Hope's translational, or forward, speed reach about 4.9 m/s (18 kph or or 11 mph) at its maximum, which happens just as her front foot plants into the dirt.  At that point, her linear momentum shifts to angular momentum as her hips and torso turn during the point of release for the ball.  The graph below shows the progression of the speed of Hope's shoulder as measured by a ground observer (click on the image for a larger view).
Hope's translational speed obviously helps increase the speed of her fastball.  The graph below shows the progression of the speed of the softball as the pitch unfolds (click on the image for a larger view).
The speed of the ball at release is about 27.8 m/s (100 kph or 62.3 mph).  The drop in speed after release is due to air resistance.  Pitch speed is further increased by cocking the wrist back, thus storing potential energy.  At the point of release, the hand flips forward like a spring, releasing some of that stored energy.

We know in physics that the net work on an object is the change in the object's kinetic energy.  Given that work is, qualitatively, a force times a displacement, increasing the distance over which a force acts is one way of increasing kinetic energy and, hence, speed.  That is the beauty of the loop-the-loop pitching motion!  Look at the photo below (click on the image for a larger view).
By executing the loop-the-loop pitching maneuver, Hope is able to work on the ball over a much larger distance than the forward distance her shoulder moves.  The yellow path in the video and in the above photo shows that the ball must undergo centripetal acceleration as part of its total acceleration.  The graph below shows the magnitude of the ball's acceleration as measured by a ground observer (click on the image for a larger view).
Coming through the bottom of her 70-cm (28-in) right arm's pendulum-like swing, the ball has an acceleration of nearly 47 times the acceleration due to gravity!  Her right arm must exert a force on the ball of over 88 N (20 lb) at the bottom of the swing, just before the release point.  That force is 25% more than the weight of the heaviest bowling ball!

During the swing of the arm, the ball rotates about Hope's right shoulder, in addition to the translational motion it has because Hope is moving forward.  The graph below shows the angular speed of the ball as measured in the rest frame of the shoulder (click on the image for a larger view).
The ball's maximum rotational speed is about 30.3 rad/s or about 289 rpm.  That rotational speed is about two-thirds the rotational speed of helicopter blades!

The last graph below shows the ball's kinetic energy during the pitch sequence (click on the image for a larger view).
The peak kinetic energy at the release point is nearly 73 J (54 foot pounds).  That size kinetic energy is not something you want hitting your unprotected head.  For comparison, a bowling ball would have that same kinetic energy if it had a speed of about 4.5 m/s (16 kph or 10 mph).  I definitely don't want a bowling ball hitting my head at 10 mph!

For all the great physics discussed above, a lot more remains untouched.  I've not gotten into the biomechanics of the arm and leg movements.  I've not discussed energy in the body, friction with the ground dirt, ball grip, and a whole host of other fun topics.  For now, watch the video again and enjoy the beautiful loop-the-loop!

21 October 2013

The Point of Center of Mass

There is a reason so many complicated problems may be analyzed with relatively simple introductory physics.  We in physics sometimes refer to the "spherical cow approximation" when simplifying complicated problems.  The main principle behind the simplification is that an object's center of mass moves under the influence of the net external force acting on the object.  So if we are not interested in all the goofy motion about the center of mass, we need only focus on the translational motion of the center of mass.  The cow is thus approximated by a point particle.

After introducing the aforementioned idea to my introductory physics class this morning, I decided to film one of my favorite objects -- a baseball bat -- in flight.  With the help of my experimentalist  colleague, Will Roach, we made the video below (click on the image for a larger view).
I marked the location of my bat's center of mass and then tossed it across our campus lawn.  The red circles and red line in the clip above show the path of the center of mass.  What does that path look like?  Check out the graph below, which shows the location of the vertical coordinate as a function of time (click on the image for a larger view).
The red circles are the data.  The blue curve is a parabolic fit to the data.  The constant vertical acceleration predicted by the fit is 9.838 m/s2 (pointing down), which is almost exactly the magnitude of the acceleration due to gravity of a point particle moving in a vacuum near Earth's surface.  Given that air resistance on my bat was not a big player in my simple toss experiment, I think the above video illustrates rather well the fact that the bat's center of mass follows the path predicted by relatively simple introductory physics.

To understand the more complicated motion about the center of mass, one must study rotations, a topic coming soon to my introductory physics course.

11 October 2013

The Power of an Axe Kick!

My last post examined a punch from my karate instructor, Mr Abercrombie.  In this post, I examine his axe kick, which involves bringing the leg up as high as possible, followed by a fast downward kick.  The knee bends very little, which makes it appear as if an axe is swinging downward.  Check out the video below (click on the image for a larger view).

Mr Abercrombie begins by executing what is called a crescent kick.  Instead of swinging across as with a crescent kick, he brings his foot straight down.  Mr Davis holds the board, which acts as victim in this demonstration.  The image below shows Mr Abercrombie's foot at the apex of its motion (click on the image for a larger view).
His foot passes through the board at about 10 m/s (36 kph or 22 mph).  His foot's acceleration through the board is nearly 15 times the acceleration due to gravity.  I included acceleration vectors in the video so that you may see how the magnitude and direction of his foot's acceleration changes during the kick.  You'll note the downward pointing acceleration as the foot rise, which is necessary to slow the foot down.  A great deal of potential energy is stored at the instant seen in the above photo.  Once he brings his foot down, the acceleration begins to turn to the left in the movie.  That is because his foot moves along the arc of a circle during the hit, which means centripetal acceleration must be present.  At the point of contact, his foot delivers about 2200 N (500 lb) force to the board, breaking it easily.

I modeled Mr Abercrombie's leg as a sequence of rigid rods.  Knowing the percentage each segment comprises of a man's body mass, I can get a reasonable estimate of the moment of inertia of Mr Abercrombie's left leg.  During the explosive kick through the board, Mr Abercrombie's leg has a rotational kinetic energy of over 200 J (almost 150 ft lb).  In the roughly one-eighth of a second it takes for his foot to go from rest at the apex to breaking the board, more than 1500 W (just over 2 hp) of power is required.  Though an athlete like Mr Abercrombie is capable of large power outputs for only a very short length of time, I am very impressed that the power output of such a kick exceeds two horsepower!

Given that the body is only about 20% efficient in its energy conversions, Mr Abercrombie probably burned about 1000 J (almost 750 ft lb) internally just to move his leg from its apex to the board.  That amount of energy represents about a quarter of a nutritional Calorie.

Did you ever imagine so much power in a kick?

09 October 2013

Karate Hits Physics of Sports!

My Physics of Sports students were treated to something special this morning.  Third-degree black belt Clifton Abercrombie and second-degree black belt Cody Davis visited my class and showed us some karate moves.  Mr Abercrombie has been one of my family's lead karate instructors for about a year-and-a-half now.  He is incredibly good at what he does.  The slow-motion video below shows one of his board-breaking feats.  He executes a perfect hammer punch, which uses the knife edge of the palm opposite the thumb.  Using the edge of one's hand allows for more force per area, i.e pressure, at the strike point.  Click on the video for a larger image.
There is a great deal of physics here!  At the start of Mr Abercrombie's motion, you will see his torso beginning to rotate.  His right arm goes back so that as his torso moves forward, he will store a great deal of potential energy in his upper arm.  Once his strong thigh, abdominal, and back muscles get his torso rotating counterclockwise (as seen from above), his right arm is like a whip.  Now cocked full of potential energy, his arm moves forward, thus turning potential energy into kinetic energy.  The red path in the above video clearly shows that after accelerating linearly in the early part of the motion, Mr Abercrombie's hand undergoes mostly centripetal acceleration as his arm follows through and breaks the board.  Note, too, Mr Abercrombie's eyes and how they are focused on his target.  As he follows through with the hammer throw, you will see his mouth open.  He yells, "Kiai!"  His exhale during the blow demonstrates proper breathing technique.

Let's now get more quantitative.  The graph below shows the speed of Mr Abercrombie's hand.  Click on the image for a larger view.
His hand's maximum speed is just over 12 m/s (43.2 kph or 26.8 mph).  That is speeding in a school zone!  To understand what accelerations are involved in making one's hand follow the path you see in the video, check out the plot below.  Click on the image for a larger view.
His hand's maximum acceleration is just over 325 m/s2, which is over 33 times the acceleration due to gravity!  A grown man's arm is nearly 6% of his body weight.  I estimate the mass of Mr Abercrombie's right arm to be 5 kg, which corresponds to a weight of about 11 lbs.  He needed about 1641 N (369 lbs) of force on his right arm to execute the hammer punch.  What's great is that he most likely used more force because my camera films at just 60 frames per second, meaning my data points are separated by 0.0167 s.  With a high-speed camera, I could get many more data points, and, I suspect, get instantaneous forces approaching twice the size of what's seen above.

What to learn to break a board like Mr Abercrombie?  Visit Super Kicks Karate in Forest, Virginia.  Click here for their Facebook page.  It's a great place to see physics in action!

30 September 2013

New Men's Marathon Record!

Kenya's Wilson Kipsang Kiprotich won the Berlin Marathon in world-record breaking fashion.  He shaved 15 seconds off the record set almost exactly two years ago in the Berlin Marathon by fellow Kenyan, Patrick Makau Musyoki (click here for my 2011 post on that).  Kipsang's time of 2h 03' 23" for the 42.195-km (26 miles 385 yards) race meant an average speed of 20.519 kph (5.700 m/s or 12.750 mph).

Each day, we see progress in humanity's athleticism.  When will someone sneak under two hours in the marathon?  When I was born in 1970, the men's record was held by Ron Hill of the UK.  His time was 2h 09' 28.8".  Now 43 years later, the record is just 06' 05.8" less time.  To get below two hours, another 03' 23" needs to be shaved off the record.  How many years had to pass for 03' 23" to be shaved down to the current record?  A time of 2h 06' 46" would have been the record by four seconds when Ethiopian Belayneh Dinsamo ran 2h 06' 50" on 17 April 1988.  Dinsamo held the record for a little more than a decade.

It took just over a quarter century to knock an amount of time off the record that will have to be knocked off the current record to see a time under two hours.  Will we have to wait another quarter century???

29 September 2013

Geno Smith and Quarterback Mechanics

About a fortnight back, I analyzed Geno Smith's three interceptions in a loss against the Patriots on Thursday, 12 September.  My work made it into a story in Metro New York, but I missed seeing it until today.  Smith has thrown two picks in each of the past two games, including today's loss at Tennessee.  To read my analysis, click here for the story by Kristian Dyer.

Soccer and Video Games

The FIFA 14 video game apparently had some bad aerodynamics in its coding.  I provided a little commentary in a recent Scientific American piece on this.  Click here for the article.  With air resistance, the shape of a kicked soccer ball's trajectory is not parabolic, which may come across that way in the article.  Anyway, it was nice to be contacted by Scientific American!

11 September 2013

US "Heads" to World Cup in 2014!

With a 2-0 win over Mexico last night, the US soccer team secured a spot in the 2014 World Cup, which will be held in Brazil.  The best Team USA has fared in a World Cup was a third place finish way back in 1930.  I hope we'll do better next summer!

Our first goal in last night's match came on a beautiful corner kick by Landon Donovan.  Eddie Johnson provided the perfect header after the Mexican goal keeper found himself a bit out of position.  Check out the somewhat fuzzy video I found for Johnson's header (click on the image below for a larger view)
What really got me excited was Donovan's kick.  Looking down on the ball while in flight, one would have seen the ball spinning counterclockwise.  I tracked the ball in the video above and you can clearly see how the ball's spin helped it curve right into Johnson's location.  The Magnus force, which is due to the air being whipped asymmetrically off the back of the ball, is responsible for the curve.  Donovan served up a banana kick with whipped air!

01 September 2013

RGIII Runs Through 1st Week of Sports Physics

Now just over a week into my first offering of Physics of Sports at Lynchburg College, I am finding that the course is more enjoyable than I had imagined (and I initially thought it was going to be loads of fun!).  At the end of the first class, I gave my students a brief survey to complete.  One of the questions asked them to list their three favorite athletes, who could be active, retired, or deceased.  One of the most popular choices was Robert Griffin III, known to most sports fans simply as RGIII (or RG3).  The dynamic young quarterback for the Washington Redskins took the NFL by storm after his Heisman-winning season at Baylor in 2011.

After an overview of dimensions and units conversions, we moved to one-dimensional motion.  London golds earned by Missy Franklin in the 200-m backstroke and Usain Bolt in the 100-m sprint made for good examples of approximately one-dimensional motion.  When it came time to analyze motion more carefully, however, I knew I had to use RGIII.  His 40-yard dash at the 2012 NFL Scouting Combine fit perfectly with what I wanted to do (click here for video of his sprint):  essentially one-dimensional motion using an athlete my students wanted to see.

The video I used was taken at 24 frames per second.  I went frame by frame and recorded RGIII's time at each yard marker.  The graph below is the result of that effort (click on the image for a larger view).
Each red data point is my best estimate of when RGIII's torso passed each yard marker (I converted yards to feet for the plot).  The blue curve is a best-fit function that helps smooth the way for time derivatives, such as RGIII's velocity, which I show below (click on the image for a larger view).
The video claims that RGIII completed the sprint in 4.38 s with a top speed of 24.6 mph (39.6 kph).  Given my admittedly rough estimations of where RGIII's torso was at each yard marker, all of which were angled from the camera view, I found that he completed the sprint in just over 4.37 s with a top speed of 23.2 mph (37.3 kph).  The NFL was certainly using more accurate timing devices, but I'm happy with how close I got.  RGIII's time and top speed are incredible!  Check out his acceleration below (click on the image for a larger view).

I have scaled his acceleration by the acceleration due to gravity.  Note that he explodes off the starting line with more than two g's!

When I began Physics of Sports, I was not planning to look at RGIII in the first week.  Letting students help decide what to cover in a sports physics course means that I get to learn things I wasn't expecting to learn.  RGIII certainly made for a fun first sporting example to analyze in detail.

11 August 2013

Cabrera vs Rivera -- AGAIN!

Just two days after the great Mariano Rivera blew his fourth save of the year when he challenged Miguel Cabrera and lost, it happened again.  Rivera was called on in the 9th inning of today's Tigers at Yankees game with a 4-2 lead.  Cabrera led off the 9th inning and sent Rivera's fourth offering 395 ft (120 m) away into the right-field seats.  With a 1-2 count, Cabrera showed why he is the game's best hitter.  Look at the image below, which shows Cabrera's dinger just leaving his bat (click on the image for a larger view).
Cabrera led with his hands, extended his arms out over the plate, and met the ball on the outside corner.  He didn't try to pull the ball, but instead went with the pitch.  Note how his hips have rotated almost completely toward the mound.  His powerful torso generated an enormous amount of torque as he moved his bat through the zone.  His head is down, his eyes are mostly on the ball, and his legs are in perfect balance.  As a physicist, I could watch Miguel Cabrera hit all day long!

So, could Mariano Rivera pick up the save with a 4-3 lead?  He got Prince Fielder to line out to third.  Victor Martinez then strode to the batter's box.  Martinez hit Rivera's second pitch 376 ft (115 m) down the right-field line for a game-tying homer.  Check out the image of Martinez connecting with Rivera's second gopher ball of the inning (click on the image for a larger view).
Note that Martinez also has his hips rotated forward.  A baseball player rotates his torso using powerful muscles in his core and upper legs.  Muscular arms won't be enough if a player's core isn't strong.  Martinez did exactly what he should have done.  The pitch was inside, and Martinez jumped on it and pulled it.  His eyes are in the right place and his front foot is a dead giveaway that he's trying to pull the ball.  Like Cabrera, Martinez took what Rivera offered and didn't try to do anything fancy.

Rivera got the next two Tigers batters out, but left the inning with his fifth blown save.  The two runs he gave up in his one inning of work took his ERA (earned run average) from 2.08 to 2.44, a more than 17% increase.

The Yankees still had the bottom of the 9th inning.  After Jose Veras retired the first two Bronx Bomers, Brett Gardner came to the plate.  On the second pitch he saw, Gardner deposited Veras's offering 393 ft (120 m) away from home plate into the second deck in right field.  Veras's walk-off homer saved the Yankees, which was reminiscent of Friday's game in which Cabrera hit one off Rivera in the 9th, only to see his Tigers lose in the 10th inning.  Take a look at Gardner's winning swing in the photo below (click on the image for a larger view).
A lead-off hitter at 5' 10" (1.78 m) tall and 185 lbs (83.9 kg mass), Gardner is not regarded as a home-run hitter.  Though just his 8th homer of the year, it was a monster!  Gardner really jumped on Veras's pitch.  His front foot shows the pull, which was the thing to do for a pitch slightly inside of center.  As with the two previous homers I discussed, Gardner's hips have rotated forward.  He may not have huge arms, but rotating his powerful core generated the necessary torque for a walk-off celebration.

Despite getting credited with his fifth blown save of the year, Mariano Rivera was given the "win" because the Yankees won the game while he was the pitcher of record in the top of the inning.  This is yet one more example of how silly the "win" stat is for a pitcher.  A "win" is a team stat, and the only thing Mariano Rivera did to help his team "win" was not give up a third run in the 9th inning.  Over the past generation, sabermetrics have given us many new stats and better ways to view the game of baseball.  As someone nearing 43 years of age, I still look at a pitcher's wins and losses because that's what I saw on the backs of the baseball cards I collected in the 1970s.  Now, I take a pitcher's win/loss record with a grain of salt.  I think Rivera would have preferred the save to the win in his stat column, though he has always impressed me as a classy gentleman who wants nothing more than to see his team win.  He may not have been happy with his performance today, but he loved seeing Gardner's ball fly over the right-field fence!

10 August 2013

Cabrera vs Rivera

My family is currently on holiday in Michigan.  We are staying at a family cottage on Lake Huron.  Being in Michigan gives me the chance to watch Tigers baseball.  Miguel Cabrera is the best hitter in baseball right now, and when he comes to bat, I stop what I'm doing and watch.

In last night's game in Yankee Stadium, the greatest closer ever, Mariano Rivera, came to the mound in the 9th inning with a 3-1 lead.  The Tigers had no chance, right?  Don Kelly pinch hit for Jose Iglesias, and hit the ball to center for out number one.  Austin Jackson doubled, and then Torii Hunter gounded out to Rivera.  That set the stage for Miguel Cabrera to face Mariano Rivera with two outs in the top of the 9th inning.  Luckily for sports fans, Rivera put the idea of an intentional walk out of his head and challenged Cabrera.

Cabrera fouled off the first pitch down the first-base line.  Yankee first baseman Lyle Overbay didn't exactly go all out to make the play, and the ball landed just out of his reach.  Had he made the catch, the game would have ended and Rivera would have had his 36th save.  Cabrera then fouled off the next pitch off his knee.  After walking around in pain, he stepped back in the box, took a ball, and then fouled a pitch off his shin.  He was pounding his own left leg!

With a 2-2 count, Yankee catcher Chris Stewart set up for a low and inside pitch.  Rivera made a serious mistake and put the ball over the plate in Cabrera's wheelhouse.  Cabrera sent the ball over the center field fence to tie the game at 3-3 and give Rivera his 4th blown save of the year.

According to ESPN's home run tracker (click here to access the site), Cabrera's home run left his bat at 105.5 mph (169.8 kph) at 24.3 degrees above the horizontal.  Reaching a maximum height of 87.0 ft (26.5 m), the ball's horizontal range was 427 ft (130 m).  Using that information, I modeled the home run.  The trajectory appears below (click on the image for a larger view).
The ball's time of flight was about 5.4 s and landed with a speed of 55.1 mph (88.7 kph).  Drag and lift are both needed to make the above plot.  Yankee Stadium is at sea level, so there was no need to correct air density for elevation.  

The Tigers lost in the 10th inning when the Yankees loaded the bases and Brett Gardner hit a two-out single past Cabrera at third.  What intrigues me is why Cabrera was playing so far in while there were two outs.  Had he been playing at normal depth, he could have fielded the ball and thrown to first to end the inning.

At least fans got to see Rivera battle Cabrera.  While writing this post, Cabrera hit another home run in today's afternoon game against the Yankees.  The guy is amazing!

02 August 2013

Aerodynamics Review Article

I was flattered to be invited by the editorial board at Sports Engineering to write a review article on aerodynamics in sport.  The article covers various ways to research aerodynamics in the sports world, and highlights recent advances made in 18 different sports.  The article may be accessed here.

Despite numerous times reading my own words, having several people read drafts of the article, having a couple of reviewers and a copy editor go through the paper, a couple of typos slipped through the cracks.  Those typos are, of course, solely my responsibility.  I hate seeing little errors slip into a publication.  Springer allowed me to publish an erratum for a misplaced decimal point.  That erratum may be accessed here.

A second typo was found by Louis Poirier, but Springer would not allow a second erratum.  The typo Louis found may be summarized as follows:  At the end of the third paragraph in Section 4.10, the distance 2.63 m was incorrectly converted to 0.802 ft.  Please replace 0.802 ft with 8.63 ft.  I thank Louis for pointing that error out to me.

If anyone notices other typos or errors, please let me know.  The paper was a lot of fun to put together, and I want it to be as accurate as possible.

21 July 2013

Kittel Bookends 100th Tour de France Won by Froome

German Marcel Kittel just edged fellow countryman André Greipel and my pick, Mark Cavendish, to win the final stage of this year's Tour de France.  Kittel took the first stage this year, as well as Stages 10 and 12.  Darkness was falling upon Paris as Kittle crossed the finish line, but the good times were just getting underway as the city will surely celebrate its 100th Tour de France well into the night.  Below is how Kittel's time compares to our prediction.
  • Stage 21:  3h 06' 14" (actual), 3h 15' 47" (prediction), 09' 33" slow (5.13% error)
I am surprised that this final stage was as fast as it was, though I shouldn't be, given how the second week went.  Below is Kittel's average speed.
  • Stage 21:  11.95 m/s (43.0 kph or 26.73 mph)
At least there was some exciting sprinting to close out this year's race!  Congratulations to Christopher Froome of Team Sky for winning this historic Tour de France.

It was a fun three weeks following the Tour de France and posting predictions.  I thank my student, Brian Ramsey, for all the work he put into this year's model.  I also thank colleagues, friends, and the many people I don't know around the world who checked out my blog.  If a little physics you found here increased your enjoyment of the Tour de France by just the tinniest amount, that's good enough for me.  I'll probably have more to write about the Tour de France in the not-too-distant future.

20 July 2013

Huge Day for Quintana!

Colombian Nairo Quintana and his Movistar Team had a monster mountain win today.  The 23-year-old climber not only dominated Mont Semnoz to reach the finish line first, he secured both the white and polka-dot jerseys AND now sits second overall, 05' 03" behind Chris Froome.  Nairo Quintana will be a major force to be reckoned with in future Tours de France!  Below is Quintana's time and the comparison with our prediction.
  • Stage 20:  3h 39' 04" (actual), 3h 45' 04" (prediction), 06' 00" slow (2.74% error)
We were worried that our time was going to be much too slow, but we had no need to worry.  We'll take another prediction under 3%!  Below is Quintana's average speed for the stage.
  • Stage 20:  9.510 m/s (34.24 kph or 21.27 mph)
Of the 170 cyclists who crossed the finish line, 25 (about 14.7%) beat our predicted time.  All the athletes should sleep well tonight after the past three days in the Alps.

The 100th Tour de France comes to an end tomorrow as Stage 21 takes riders between two famous cities.  Beginning in Versailles, cyclists loop south for a couple of short category-4 climbs before heading northeast to Paris.  The flat stage has a length of 133.5 km (82.95 mi).  The ride along the famed Avenue des Champs-Élysées toward the Arc de Triomphe will be a day to remember as the 100th edition of the world's most famous bicycle race reaches its climax.  Chris Froome should close the deal and give Team Sky two wins in a row.

The final stage is difficult to predict because so much of it is ceremonial.  There are some good sprints, and Mark Cavendish usually dazzles, but predicting how competitive the stage will be is tough.  Nonetheless, we offer our final prediction for this year's Tour de France below.
  • Stage 21:  3h 15' 47" (prediction)
That time might be a bit fast, but we are hoping to see some great racing in addition to all the pomp and circumstance.  Enjoy the final stage!

19 July 2013

Costa Takes Stage 19 in the Rain!

Rui Costa of Portugal won his second mountain stage today.  There was a lot of rain in the latter part of the route.  We were denied a great downhill sprint to finish the stage.  Cyclists were cautious on the final descent, including having to dodge some nutty fans who were out in the road with their umbrellas.  Below is Costa's winning time and the comparison with our prediction.
  • Stage 19:  5h 59' 01" (actual), 5h 48' 01" (prediction), 11' 00" fast (-3.06% fast)
We like our small error and, as I wrote yesterday, we were a faster than today's top cyclist.  Below is Costa's average speed.
  • Stage 19:  9.494 m/s (34.18 kph or 21.24 mph)
I mentioned yesterday that I thought the winning time should be over six hours.  When I saw all the rain on the final descent and watched Costa cruise to victory with no real sprint at the end, I was even more surprised that the top two cyclists sneaked in under six hours.  The reason is that today's stage is almost an exact replica of Stage 17 in the 2004 Tour de France.  Click here and scroll down to Stage 17's profile.  When you compare that profile with today's profile (click here, scroll down, and click on "Stage profile"), you'll see the same 204.5-km distance, the same two early climbs to Col du Glandon and Col de la Madeleine, and then a category-2 climb followed by two category-1 climbs.  The second category-1 climb was indeed a different mountain in 2004 compared to today's stage, and there are some more differences in the middle of each stage, including the location of the sprint.  I am neither claiming that the two stages are exact nor that cycling Stage 17 in 2004 is exactly the same as cycling Stage 19 in 2013.  But, the stages are about as similar as two stages of the Tour de France can be:  same beginning and ending communes (sightly different elevations), same distance, and almost the same climbs.  Lance Armstrong won 2004's Stage 17 with a time of 6h 11' 52", which is why we thought our prediction was going to be much too fast today.  We are impressed to see 46 of 170 (about 27%) cyclists finish today's rainy stage in a time less than 6h 11' 52".

Given that they finished today's stage in the same group, Froome maintained his 05' 11" lead on Contador.  It is clearly Christopher Froome's Tour de France to lose.

Tomorrow's final mountain stage is short at just 125 km (77.7 mi).  Beginning in the commune of Annecy, Stage 20 has cyclists tackle a category-2 climb early on, followed by three category-3 climbs by the time they reach the two-fifths point of the route.  They then have a category-1 climb to the 1463-m (4800-ft) peak of Mont Revard.  The Alps bids cyclists adieu with an hors catégorie stage-ending climb to the 1655-m (5430-ft) peak at Mont Semnoz.  As much as we thought today's stage prediction would be too fast, we think that the way this year's Tour de France has played out, we'll return to being too slow tomorrow.  Below is our Stage 20 prediction.
  • Stage 20:  3h 45' 04" (prediction)
If I were betting, this year's Tour de France makes me think 3h 15' 00" will be more likely tomorrow.  But, if the past two, grueling mountain stages have worn the cyclists down a little, we might be have a shot.  Let's hope there is no rain tomorrow!

18 July 2013

A Great Day for France!

What a phenomenal stage today!  My student, Brian Ramsey, and I sat in my office and watched the second half of today's stage with its historic double pass on Alpe-d'Huez.  Right at the 2-km (1.2-mi) mark, France's very own Christophe Riblon found a reserve of energy and kicked it into high gear.  Riblon blew past Tejay van Garderen of the US and then savored the final few hundred meters of the stage as throngs of French cheered him on.  Below is how our prediction fared against Riblon's great day.
  • Stage 18:  4h 51' 32" (actual), 4h 59' 50" (prediction), 08' 18" slow (2.85% error)
We are extremely pleased to hit such an unpredictable and challenging stage to better than 3%!  Of the 175 cyclists who finished today's stage, 32 (about 18%) beat our predicted time today.  Riblon's average speed is given below.
  • Stage 18:  9.862 m/s (35.50 kph or 22.06 mph)
What an impressive ride for Riblon!  I hope cyclists are able to get plenty of rest tonight.  Tomorrow's 204.5-km (127.1-mi) mountain stage will require lots of energy expenditure, perhaps as much as 9000 Calories.  Beginning in the commune of Le Bourg-d'Oisans, cyclists will be immediately hit with an hors catégorie climb to reach the 1924-m (6312-ft) peak of Col du Glandon.  A lightening-fast downhill leads to another hors catégorie climb to Col de la Madeleine's peak at an even 2000 m (6562 ft) elevation.  A category-2 and two category-1 climbs must be traversed before reaching the stage's end north of the starting point at the commune of Le Grand-Bornand.  Below is our prediction.
  • Stage 19:  5h 48' 01" (prediction)
Of all this year's Tour de France stages, this is the one we thought our model would get too fast.  I'll explain in tomorrow's post why I think this stage should be won in a time that goes over six hours.  If tomorrow's winner sneaks in under six hours, I'll be writing about top speeds again.

Froome gained nearly a minute of time today over Contador for the overall classification lead.  Will anyone catch Froome tomorrow???

17 July 2013

Froome Edges Contador in Rainy Time Trial!

Chris Froome was the last rider in, and just eclipsed Alberto Contador's time by nine seconds to win today's individual time trial.  Contador is now second overall at 04' 34" behind Froome.  Below is how Froome did against our prediction.
  • Stage 17:  51' 33" (actual), 44' 49" (prediction), 06' 44" fast (-13.06% error)
Our prediction for this time trial now represents our worst prediction of the race.  After the way this Tour de France has gone, we would have been shocked that we were so fast with our prediction had we not observed the competition.  But today was not an ordinary race day.  There were thunderstorms, swirling winds, and even hail at times.  Riders slowed through some very dangerous spots, especially on the two major downhills.  Instead of seeing possible through-the-roof speeds, Mother Nature slowed everyone down.  Even the Tour de France's time-schedule conservative estimate of 47' was well off the winning time.  Just when we thought we had a predicted time that could compete with this year's speedy cyclists, the weather put a stop to that!  Froome's average speed is below.
  • Stage 17:  10.35 m/s (37.25 kph or 23.14 mph)
Probably the most anticipated stage of this year's Tour de France is tomorrow's Stage 18.  The 172.5-km (107.2-mi) long mountain stage begins where Stage 16 left off, in the southeastern French commune of Gap.  The stage ends north of Gap at the famous ski resort Alpe-d'Huez, which sits at an elevation of 1850 m (6070 ft).  Alpe-d'Huez has appeared in several Tours de France, always challenging riders with its renowned 21 hairpin turns (click here for a great Wikipedia image).  What makes this year's Stage 18 so special is that for the first time in the 100-year history of the Tour de France, the ascent to Alpe-d'Huez will be made twice.  Cyclists will reach Le Bourg-d'Oisans after being on their bikes for 108 km (67.1 mi) and having already navigated a category-3 climb and two category-2 climbs.  They will then begin their first climb toward Alpe-d'Huez, an hors catégorie climb for sure.  Reaching the peak won't be their high-elevation mark for the day because they then have a category-2 climb to the top of Col de Sarenne, which sits at 1999 m (6558 ft) above sea level.  A sure-to-be fantastic downhill sets 
up the final hors catégorie return to Alpe-d'Huez.

When we developed our model for this year's Tour de France, we were especially interested in what our model had to say about Stage 18.  Below is our prediction.
  • Stage 18:  4h 59' 50" (prediction)
So many unpredictable complications take place on mountain stages.  The weather is unpredictable, elevation changes wreak havoc on cyclists, and strategies may change on the fly.  Regardless of how our prediction comes out, tomorrow's stage is bound to be memorable!

16 July 2013

Costa Takes Stage 16!

Portuguese cyclist Rui Costa won today's Stage 16.  He cruised to the finish line with a 42-second lead over the next rider in.  Below is Costa's time compared to our prediction.
  • Stage 16:  3h 52' 45" (actual), 4h 12' 20" (prediction), 19' 35" slow (8.41% error)
As advertised yesterday, we once again came in too slow.  My "throw my hands in the air" guess of 3h 45' 00" would have made for a fine prediction today.  Today's race began with a temperature around 32 C (89.6 F) and a 10 kph (6.2 mph) tailwind.  I knew our prediction was in trouble when I saw the tailwind.  Winds then swirled for much of the second half of the stage.  Below is Costa's average speed.
  • Stage 16:  12.03 m/s (43.3 kph or 26.91 mph)
We don't find that average speed to be especially anomalous.  With early tailwinds and higher-than-expected overall speeds this year, our prediction is about where we thought it would be.  Of the 179 riders who competed today, 93 (about 52%) beat our predicted time.  In 2003, getting a prediction under 10% was the expectation.  A decade later, we expect to do much better.

To give you a feeling of where things stand after 16 stages, we note that Chris Froome, who retains the yellow jersey after today's stage and now has a 04' 14" lead over Bauke Mollema of the Netherlands, currently has an overall average speed of 41.92 kph (26.05 mph).  After 16 stages (i.e. Stages 0-15) in last year's Tour de France, Bradley Wiggins was the overall leader with an average speed of 40.27 kph (25.02 mph).  If the 2012 version of Bradley Wiggins were competing right now with that same average speed, he would currently sit in 165th place with just 15 cyclists behind him.  Clearly, the first 16 stages of this year's Tour de France are different from the first 16 stages of last year's race.  What is written in this paragraph is not meant to be a rigorous comparison between the race after 16 stages this year and the first 16 stages from last year.  I offer the average speed comparison merely to point out that speeds are up this year.  I'll make the same comparison once this year's Tour de France ends.

Tomorrow's Stage 17 is an individual time trial of length 32 km (20 mi).  Beginning in the southeastern French commune of Embrun, the stage heads due west to the commune of Chorges.  The time trial contains two category-2 climbs.  Below is our prediction.
  • Stage 17:  44' 49" (prediction)
The way the second half of this Tour de France is going, we fully expect to see jaw-dropping speeds tomorrow.

15 July 2013

Stage 16 Prediction

The Tour de France picks up tomorrow in the southeastern French commune of Vaison-la-Romaine.  The medium mountain stage takes riders 168 km (104 mi) east and slightly north to the commune of Gap.   Riders meet a category-3 climb early, followed quickly by a category-2 climb.  A second category-2 climb in the Alps awaits riders near the stage's end as they'll ascend the 1268-m (4160-ft) peak of Col de Manse.  The stage should end with a great downhill sprint.  Cyclists will meet Col de Manse again on Thursday in this year's much-anticipated Stage 18.  Below is our Stage 16 prediction.
  • Stage 16:  4h 12' 20" (prediction)
I fully expect our prediction to be slow once again.  The temptation to tweak our model in the middle of the race is strong, but I resist that because such a tweak at this point is not scientific.  We need more time to explore motivations for needed changes.  If I were betting on tomorrow's stage, I might go with a time of 3h 45' 00", given how fast the cyclists were in the past week.  As I've already noted, our model that worked well in 2012 worked well during the first week of this year's race, but did not work well last week.  There is much to learn this week!

14 July 2013

Froome is King of the Mountain!

Chris Froome gave Team Sky a huge win today after beating Nairo Quintana of Columbia to the top of Mont Ventoux.  Our prediction was just terrible.
  • Stage 15:  5h 48' 45" (actual), 6h 30' 17" (prediction), 41' 32" slow (11.91% error)
This stage is, by far, our worst predicted stage of this year's Tour de France.  Below is Froome's unbelievable average speed for today's win.
  • Stage 15:  11.60 m/s (41.72 kph or 25.92 mph)
To give you some idea of why we are stunned by what has been happening in the past week, consider a few more details about today's stage.
  1. As I wrote yesterday, today's monster climb began after 221 km (137 mi) of cycling.  When the lead cyclists reached that point today, they were averaging just a hair under 47 kph (29 mph).
  2. Knowing that climbing Mont Ventoux was ahead of them, cyclists had biked 30 km (19 mi) longer than yesterday's Stage 14, and the leaders had an average speed 2 kph (1.2 mph) greater than yesterday's winners.  Granted, yesterday's stage had more climbs than the first 221 km of today's stage.  Still, that is quite a pace for a distance that exceeds all but two stage lengths this year.
  3. My estimate is that Froome finished at 4:33 pm local time.  The Tour de France's website created three time schedules based on three average speed estimates.  Those estimates were, from high to low, 39 kph, 37 kph, and 35 kph.  The time schedule has the finish time for the fast average speed estimate at 4:58 pm local time, 6h 23' after the race started, and 5:41 pm for the slow average speed estimate, 7h 06' after the race started.  As I've noted in my last couple of posts, I hope those looking for live action in France are getting to various points along the route well in advance of the time schedule estimates.
  4. When Juan Manuel Gárate of Spain won Stage 20 in the 2009 Tour de France, which was a 167-km (104-mi) long stage that ended atop Mont Ventoux, his average speed was 35.87 kph (22.29 mph).
  5. When Richard Virenque of France won Stage 14 in the 2002 Tour de France, which was a 221-km (137-mi) long stage that ended atop Mont Ventoux, his average speed was 38.61 kph (23.99 mph).  Froome biked 21.5 km (13.4 mi) farther today, and it took him just five minutes longer than the time Virenque needed in 2002.
  6. When Marco Pantani of Italy won Stage 12 of the 2000 Tour de France, which was just 149 km (92.6 mi) long and also ended atop Mont Ventoux, his average speed was 35.03 kph (21.77 mph).
I joked with a few colleagues yesterday that if today's winner had an average speed over 40 kph, I should jump out my office window and see if I, too, can fly.  To see nearly 42 kph as the winner's average speed for this stage boggles my mind.  180 cyclists beat our time today; one did not.  Froome clearly retains the yellow jersey as well as the polka-dot jersey.

What has happened in the Tour de France this past week has been unprecedented.  Not only do the cyclists need a day off tomorrow, I need a day off to get my head to stop spinning.

Our sure-to-be-slow prediction for Stage 16 will be posted tomorrow.

13 July 2013

Trentin Wins Stage 14!

Matteo Trentin ended Italy's three-year absence from the winner's podium by taking today's Stage 14.  No shock how our prediction fared against this year's fast cyclists.
  • Stage 14:  4h 15' 11" (actual), 4h 35' 15", 20' 04" slow (7.86% error)
As with Stage 13, my "if they're still flying more than in 2012" prediction of 4h 20' 00" was still five minutes too slow.  Also as in Stage 13, all 181 cyclists beat our time.  We went from great predictions early on to rather slow predictions for the past four stages.  Trentin's average speed is given below.
  • Stage 14:  12.47 m/s (44.91 kph or 27.91 mph)
Today's stage was 191 km long.  How fast is the above speed?  Stages 2 (156 km) and 3 (145.5 km) from this year's race were both classified medium mountain.  The averages speeds for the winners in those two stages were 41.94 kph and 39.43 kph.  The Tour de France's website had 43 kph as its maximum estimate for today's Stage 14 (I hope those of you in France got out earlier than what was on the time schedule!).  Today's last place finisher (181st place) had an average speed of 43.02 kph.  How about a comparison to last year's four medium-mountain stages' winners' average speeds?  Check out the table below (2012 Tour de France medium-mountain stages).

Stage Length (km) vave (kph)
3 197.0 41.77
7 199.0 39.99
8 157.5 40.01
12 226.0 39.56

Medium-mountain and mountain stages are probably harder to compare from year to year than flat stages because weather, strategy, and all the random things that happen during a stage can make comparisons in different mountains moot (i.e. simply looking at length and average speed is not good enough for a scientific comparison).  With much study ahead of us, all I can do now is a superficial comparison.  Today's last-place finisher's average speed beat the winners of this year's Stages 2 and 3 and all of last year's medium-mountain stage winners.  To say that we are perplexed by what is happening in the middle of this year's Tour de France is an understatement.

Tomorrow's Stage 15 is a 242.5-km (150.7-mi) mountain stage.  When the stages were released for this year's Tour de France, I thought Stage 15 looked incredible.  Imagine what it takes to cycle 221 km (137 mi), enduring three category-4 climbs and one category-3 climb along the way.  Think you would be tired?  That 221-km distance is longer than all but two of this year's 21 stages.  So, imagine beginning in eastern France in the commune of Givors and then biking for 221 km all the way to the commune of Bédoin, which sits in southeastern France.  You would have biked a distance that would have had you finished in 19 of 21 stages this year.  But you wouldn't be finished yet!  Another 21.5 km (13.4 mi) remains in the stage.  You reach Bédoin at an elevation of 330 m (1083 ft) knowing you won't be finished until you ascend Mount Ventoux and reach an elevation of 1912 m (6273 ft or 1.188 mi).  How do you fancy that?!?  The hors catégorie climb is a 7.5% grade on average and will surely separate contenders from pretenders.

Given how the middle of this race has gone, the prediction below is offered with tepid confidence.
  • Stage 15:  6h 30' 17" (prediction)
There is a reason that Monday is a rest day.  Enjoy the magnificent scenery and the grueling 1582-m (5190-ft) climb up the "Beast of Provence" in the final 21.5 km (13.4 mi) of tomorrow's stage!

12 July 2013

The Manx Missile Blasts Off Again!

Mark Cavendish won his second stage of this year's Tour de France.  Chris Froome maintains his hold on the yellow jersey.  As I expected, we were once again too slow.  Below is Cavendish's time compared to our prediction.
  • Stage 13:  3h 40' 08" (actual), 4h 01' 57" (prediction), 21' 49" slow (9.91% error)
Despite 15 kph (9.3 mph) crosswinds with headwind components, Cavendish was able to achieve the average speed you see below.
  • Stage 13:  13.10 m/s (47.15 kph or 29.30 mph)
Check out my post from yesterday.  Granted, only Stage 15 from last year's race was shorter than today's 173-km (107-mi) stage.  Still, no average speed from last year's flat stages comes close to what was seen today.  Last year's Stage 15 was 14.5 km (9.01 mi) shorter than today's stage, yet the winner's average speed in that stage was 43.18 kph (26.83 mph).  Even my super-low estimate of 3h 45' 00" at the end of yesterday's post would have been five minutes too slow.

All 181 cyclists beat our prediction today.  The last rider's time was 3h 53' 44", which gives an average speed of 44.41 kph (27.59 mph).  That would have been the third fastest average speed for last year's flat stages.  Put another way, today's last-place finisher had an average speed that would have won six of last year's flat stages.

To say we are flummoxed by the speeds we've seen in recent stages would be an understatement!  For those of you in France who are watching the race live, be sure to get out earlier than what the Tour de France website suggests on its time schedule.  Today's time schedule had 46.0 kph (28.6 mph) for its maximum estimated average speed.

Tomorrow's Stage 14 is of the medium-mountain variety.  The 191-km (119-mi) stage begins in the commune of Saint-Pourçain-sur-Sioule and ends to the southeast in the city of Lyon.  Along the way, cyclists will meet five category-4 climbs and two category-3 climbs.  Our prediction is given below.
  • Stage 14:  4h 35' 15" (prediction)
If this were 2012, I would be quite confident with the above prediction.  Given how this year is going, I won't be surprised to see something close to 4h 20' 00".  I am not trying to play two predictions.  We are sticking with what is written above because we want to be fair to the model we created when this race started.  It is clear, however, that the jump in athletic performance and technology from last year to his year is greater than we anticipated.

11 July 2013

Kittel wins his third stage!

German Marcel Kittel won his third stage of this year's Tour de France (recall he won Stages 1 and 10) by just barely beating out Mark Cavendish at the finish line.  Chris Froome still holds the yellow jersey.  Below is how Kittel's time compared to our prediction.
  • Stage 12:  4h 49' 49" (actual), 5h 02' 54" (prediction), 13' 05" slow (4.51% error)
There is a clear theme with our modeling this year.  We simply do not have quite enough power output to match the winning times we are seeing.  All 182 cyclists beat our prediction today, and that was despite 20 kph (12 mph) crosswinds with predominately headwind components.  Check out the average speed below for Kittel's win.
  • Stage 12:  12.54 m/s (45.13 kph or 28.04 mph)
You see the average speed above for today's 218-km (135-mi) flat stage.  Below is a table of last year's (2012 Tour de France) flat stages and the average speeds for the winners of those stages.

Stage Length (km) vave (kph)
1 198.0 39.82
2 207.5 41.92
4 214.5 40.40
5 196.5 41.88
6 207.5 44.95
13 217.0 43.69
15 158.5 43.18
18 222.5 45.38

I left out the ceremonial last stage.  Only Mark Cavendish's phenomenal ride in last year's Stage 18, a stage on which we were 7.10% slow, has an average speed in excess of today's average speed.  What is all the more remarkable about Cavendish's win in last year's Stage 18 is that that stage was 4.5 km (2.8 mi) longer than today's Stage 12.  Whether last year's flat stages were longer or shorter than today's stage, only last year's Stage 18 exceeds today's speed.

It should be obvious from the above table why we have been surprised by this year's race.  The technology that goes into the bikes and apparel, the teams' strategies, and the athletes' performances must have improved considerably since last year's race.  Every Tour de France is different and every stage is different.  Still, we are impressed with this year's speeds.

Tomorrow's Stage 13 is also flat.  It picks up where today's left off, in Tours, and sends riders 173 km (107 mi) southeast to the commune of Saint-Amand-Montrond, which is located right in the center of France.  Below is our prediction.
  • Stage 13:  4h 01' 57" (prediction)
Will riders save a little for the daunting mountain stages in the Alps?  Or will they go all out to secure the best time position before the climbing begins?  There is a short category-4 climb in tomorrow's stage, but the route is mostly flat.  Given how fast cyclists have been this year, I won't be surprised to see a winning time closer to 3h 45' 00", but I'll stick with the prediction above.