Tim Cahill: Goal of the Tournament!

"It's the goal of the tournament!"  My summer research student, Chad Hobson, uttered those words immediately after seeing Australia's Tim Cahill strike as incredible a goal as you are ever to see.  We were watching Netherlands take on Australia in my office,  hoping not only to see some great soccer, but some great physics.  We weren't disappointed as the quality of play was top notch, and there was plenty to dissect from a physics viewpoint.

Just a minute before Cahill's great goal, Holland's Arjen Robben scored an incredible goal of his own.  Firing with his left boot from the left side of the penalty box, he sent Brazuca through the Australian defense as one would thread a needle.  Check out Robben's shot below (click on the image for a larger view).

It appears Australia has a great chance to block the shot, right?  Well, watch Brazuca sneak under the first defender's boot (click on the image below for a larger view).

The ball had only the goal keeper to beat, which it did (click on the image below for a larger view).

What a shot!  We could hardly believe what we had just seen, and then Tim Cahill struck the equalizer.  What's great about the start of the play is the Ryan McGowan met a pass moving almost perpendicular to his run.  He in fact kicked the ball without a single dribble (click on the image below for a larger view).

Not only does McGowan kick the ball just 15 yards (13.7 m) from the halfway line, he is kicking the ball to Cahill (in yellow on the far right above) who will have to catch it while looking into the sun.  Look at the stadium shadow on the pitch!  When thinking about what Cahill did, don't forget that he was staring into the sun.  The view below shows that Cahill was about 9 yards (8.2 m) from the penalty box when McGowan kicked the ball (click on the image for a larger view).

The straight-line distance from McGowan's kick to the point where Cahill catches the ball with his left boot was about 41 yards (37 m).  The ball took a bit more than 2.7 seconds to reach Cahill, and Cahill struck the ball immediately upon receiving it.  He never dribbled the ball; the ball never touched the pitch.  The image below shows Cahill receiving the ball (click on the image for a larger view).

The timing was exquisite!  McGowan was running about 15 mph (24 kph) when he kicked the ball.  He launched it off his boot with a speed of approximately 51 mph (82 kph).  Cahill was running forward at roughly 11 mph (18 kph) when he caught/kicked the ball, which was around 2 feet (0.6 m) off the pitch.  The ball was traveling about 37 mph (60 kph) when it hit Cahill's left boot.  The image below shows a better look at the catch/kick (click on the image for a larger view).

Note the players' shadows.  The ball came right out of the sun from Cahill's perspective.  He kicked Brazuca into the middle of the top crossbar, which then ricocheted into the goal.

If you haven't seen this goal, go online and find it.  It's worth watching!

USA Corners Its Way to Win!

Two days have passed since I was jumping up and down after my home country defeated Ghana, 2-1.  Ghana has had our number in recent World Cups, so a win was sweet, and even more so given the quality of competition in our group.  The Wall Street Journal published today a little blurb I sent them on the corner kick that set up the winning goal.  I've only now had time to add a few more comments beyond what the Wall Street Journal published.

About four minutes following a great equalizing goal by Ghana in the 82nd minute, USA had a corner kick right of the goal as seen by someone looking at the goal from the middle of the pitch.  The image below shows Graham Zusi in the 86th minute about to deliver the corner kick (click on the image for a larger view).

He kicked the ball with his right boot, which imparted counterclockwise spin (as seen from above) on the ball.  He launched the ball at a speed of about 59 mph (95 kph).  The ball curved to his left, creating a perfect banana trajectory.  John Brooks headed the ball into the goal after the ball spent 1.47 seconds in the air.  The image below shows Brooks with Brazuca on his head (click on the image for a larger view).

The magnitude of the force Brooks exerted on the ball, which is the same magnitude force he felt from the ball, was nearly 120 pounds!  As great as Brooks's header was, I really love the corner kick that set it up.  Check out the trajectory below (click on the image for a larger view).

The solid red curve is the actual trajectory.  The red dashed curve shows the shadow on the pitch.  The dark thin curve is what the trajectory would have looked like has there been no spin on the ball, i.e. no banana kick made from whipped air using the Magnus force (that kick's shadow is also shown).  The real ball moved nearly 6 yards (5.5 m) laterally to the left.  Moving toward Brooks helped him with his header.  He could give the ball speed into the goal by almost reversing the normal component of the ball's impact velocity.

Isn't that corner kick a thing of beauty?

Torque that Brazuca!

The first two goals in Switzerland's 2-1 win over Ecuador gave us great banana kicks from the left side into the penalty area.  The assists were every bit as spectacular as the headers that provided the goals.  Both relied on left-booted kicks that put clockwise spin (as seen from above) on the ball.  The left-to-right curve due to the the Magnus force meant the ball had a component of its velocity away from the goal and toward the waiting head of a teammate.  Both kicks were things of beauty!

Ecuador's Walter Ayoví kicked the ball by swinging his left leg across and upward.  Look at the image below (click on image for a larger view).

What a great image!  Notice the body lean.  The planted right boot pushed on the pitch to the right (as we see it in the image).  The pitch, by Newton's third law, pushed the right boot to our left.  That created a torque in the clockwise direction (again, as seen by us), which nicely balanced the counterclockwise torque created by Ayoví's weight.  That weight torque was clearly present because Ayoví's center of mass was not over his feet.  Enner Valencia was set up perfectly.

By swinging his left boot across Brazuca, Ayoví created the necessary torque to get the ball spinning.  That same type of torque was put on Brazuca by Ricardo Rodriguez of Switzerland as he sent the ball into the penalty area on a corner kick so that Admir Mehmedi could head in the equalizer.  I cropped the image below from my video feed (click on image for a larger view).

The net partially obstructs the view, but you should be able to see the body lean and Brazuca coming off Rodriguez's left boot.  Like the previous kick, boot must meet ball left of center as seen by the player.  A nonzero lever arm distance from the contact point to Brazuca's center of mass is necessary to get the ball spinning.

I can't leave this Sunday of World Cup action without commenting on the incomparable Lionel Messi.  After incredible footwork to get to one of his favorite spots on the pitch, Messi took advantage of an opening to send the ball off the left post and into the goal.  Look at the image below (click on image for a larger view).

Another left boot, but this time the line of force from the boot went above Brazuca's center of mass.  Messi's left boot is moving upwards in the above image.  Messi's genius was on full display not only in getting to the perfect spot for a shot on goal, but knowing to give Brazuca topspin.  Messi kicked the ball from 18 yards out, right on the center of the line for the penalty area.  The ball stayed low, and bounced just before hitting the left post.  There was no way the goal keeper could get such a shot.  The topspin meant there was a Magnus force pointing down, which helped keep the ball low to the pitch.  Watch that goal, and be amazed at one of the best to ever play the beautiful game.

To Spin or Not to Spin?

The two goals scored in the first half of Italy's match with England were great, both from a strategic point of view and from physics point of view.  Claudio Marchisio scored the match's first goal after Italy executed a perfect corner kick play.  Instead of powering the ball into the penalty area, Marchisio received the ball after some fancy and tricky footwork from teammates.  He then kick the ball low to the pitch with almost not spin.  Check out the image below (click on image for larger view).

The image I cropped from the video feed I was watching shows the ball low to the pitch with very little spin.  Note how clear the ball is compared to the players.  Marchisio's shot looked to have a slight knuckle effect, which I estimate from one replay that showed the shot from the back of the goal.  At the speed and angle of the launch, the ball would not have laterally deflected more than about 2% of its horizontal range (from boot to the first time the ball hit the pitch).  He opted for no spin because he was after a straight shot on goal.

England tied the match just two minutes later when Wayne Rooney found Daniel Sturridge with a beautiful cross off his left boot.  The ball had lots of clockwise spin (as seen from above), which meant it curved left to right as Rooney would have seen it.  Look at the image below (click on image for a larger view).

Rooney is off to the right, out of frame of the video feed I cropped.  His perfectly placed ball is heading for Sturridge.  The ball's spin meant that it arced a little toward Sturridge, which made for easy handling by the forward.

The first half goals showed when it's good to kick with little-to-no spin and when a lot of spin is needed.  Great physics on those two goals!

Italy got the winning goal on a well-timed header by Mario Balotelli just five minutes into the second half.  An excruciatingly close miss by Rooney and several errant kicks by England late helped Italy close out a 2-1 win and three very big points on the heels of Costa Rica's stunning win over Uruguay.

Great Header, Robin van Persie!

What a great goal by Robin van Persie just before halftime of the Netherlands match against Spain!  Daley Blind fired the ball off his left boot from just past the middle of the pitch near the left touch line.  The ball had lots of clockwise spin (as seen from above), and curved left to right.  The ball met the head of a diving van Persie about 16 yards from the center of the goal.  The trajectory of the kick was perfect, and so was van Persie's header, which sent the ball upward and over the Spanish goal keeper.  Is the image below great, or what?!?

Click on the above image for a larger view.  Spain is in trouble as Arjen Robben just used some fancy footwork to score in the 53rd minute.  This great game is better than the rain-soaked match between Mexico and Cameroon that saw two goals taken from Mexico in the first half by truly awful refereeing.

Almost World Cup Time!

Like millions and millions of other people around the world, I'm anxiously awaiting for tomorrow to bring the start of the World Cup in Brazil.  I've had a lot of fun researching the new ball, which is called Brazuca.  That research, getting ready for the Tour de France, and working on a couple of books have kept me very busy -- and away from blogging.  I hope to get a few posts written during the World Cup.

My research student this summer is Chad Hobson, who just completed his first year here at Lynchburg College.  Chad will be helping me with Tour de France modeling.  He is also a soccer player and avid fan of the beautiful game.  We spent part of the afternoon today examining the flight of Brazuca.  The photo below shows my ball, which is just like the one my colleagues in Japan tested in a wind tunnel (we published our research on Brazuca a couple of months ago -- go here to download a copy of our paper for free).

Click on the image for a larger view.  Note the texturing on the surface.  Note, too, the long seam paths, which help make the six-panel Brazuca's surface rougher on average than the eight-panel Jabulani surface.  The research I did with my colleagues in Japan shows that Brazuca will be more stable than Jabulani.  There should not be so many erratic trajectories this time around!

To get a feeling for how a soccer ball curves, check out the image below (click for a larger view).

Chad kicked the ball with counterclockwise spin as seen from above (also a little topspin component).  I show the path the ball takes with the red trajectory curve.  Brazuca's spin causes air to be whipped off back and to the right.  There is thus a leftward deflection, creating the nice banana trajectory you see above.

It now World Cup time!

Brazuca Physics

During the past couple of months, a few colleagues have contacted me about my recent dearth of blog writing.  Though trite to offer, I've been busy!  Research work, teaching, and administrative duties have made little time for blog writing.  What got me motivated to write a post today was an invitation from NPR to appear on All Things Considered yesterday.  I was interviewed by Melissa Block about the Brazuca, which is the name of the new World Cup ball that will be employed in this summer's world tournament in Brazil.  Click here to get to a page where you may listen to my four-minute interview.

In late 2013, I was invited by Takeshi Asai of the University of Tsukuba in Japan to collaborate with him on aerodynamics research of the Brazuca.  Dr Asai and his colleague, Sungchan Hong, supplied me with wind-tunnel data for the Brazuca, as well as wind-tunnel data for the Jabulani, which was used in the 2010 World Cup.  I spent the majority of my winter "break" analyzing data, coding trajectory models, and paper writing -- all LOTS of fun and why I LOVE my job!  My colleagues and I came to the conclusion that the Brazuca is a better ball than the Jabulani.  Much of the wild knuckling effects seen in 2010 should be absent this summer.  The key is that despite six panels on the Brazuca compared to eight on the Jabulani, the total seam length on the Brazuca is 68% longer.  That added roughness leads to a drag crisis at a smaller speed, meaning most intermediate-speed and all high-speed kicks should be in the post-critical region where the drag coefficient is essentially uniform.  The Brazuca also shows better stability than the Jabulani due to more uniformity in the surface roughness.  Many more details appear in our paper, which was recently published in the Journal of Sports Engineering and Technology (click here for a link to the paper).

There are only 50 more days before the 2014 World Cup commences.  I'm anxious to see the new ball in action!

The Biathlon is Cool!

On today's Winter Olympics and physics radio spot at The Takeaway, I talk about the biathlon.  Click here to get to the page with the "listen" link.  How great is the biathlon, which is gift from Norway?  If you like skiing and shooting, there is no better sport!

Winter Olympics Radio Links

The Takeaway combined this week's radio spots into a single download.  They may be accessed here.  There should be a new segment on the biathlon next Monday.

This past Wednesday, I was on NPR's All Things Considered.  Most of the conversation concerned ski jumping.  Click here for a brief write-up and a download link for my segment of the show.

It's fun chatting about the Winter Olympics -- so much great physics involved!

Speed Skating!

On today's Winter Olympics radio show for The Takeaway, I talk about speed skating.  Click here for the link.  Lots of great physics, including Newton's laws and angular momentum conservation, dominate the conversation.

Freestyle Skiing and Ski Jumping

At The Takeaway today, I talk about freestyle skiing and ski jumping.  Click here for the story and a link to the audio from the radio show.  There is fascinating physics as one converts gravitational potential energy into kinetic energy, all the while either fighting the air (drag) or getting help from the air (lift).  What's great about the ski jump this year is that women finally get to compete in that Olympics event!

Ski Jumping and Thoughts on Science and Sports

I worked on a piece for the New York Times on ski jumping.  The graphics are great!  Click here for the story.  It's about time women get to compete for Olympic gold in this event!  There is so much wonderful physics in ski jumping.  The "V" style alone has increased lift by about 30% over the old skies-pointed-forward style.  Look for some birds that spread their feathers while in flight, increasing lift just like the separated skies of a ski jumper.  We may learn a lot from evolution!

I was invited to write a short blog post for the Johns Hopkins University Press.  The topic is science and the Winter Olympics.  Click here for the post.  In no way does understanding some science behind sports diminish the viewing experience.  There is so much beauty in the laws of nature, and I marvel at how human beings continually push the envelope of what's possible.  The athletes are special, but so are the scientists who design better and better equipment, craft more sophisticated training regimen, and use modeling to find seemingly small ways that improve performance.  For events timed to the nearest thousandth of a second, tiny improvements may mean the difference between seeing one's country's flag from the podium and watching from the stands.

Winter Olympics Chat on The Takeaway

I had a lot of fun chatting with John Hockenberry for his radio show at The Takeaway.  We talked about several sports featured in the Sochi Winter Olympics.  The interview will be chopped into a few segments, beginning with curling and speed skating.  Click here for a preview.  If you are in one of the more than 220 cities where The Takeaway is syndicated, tune in for some physics fun!  I believe there will also be podcast episodes on iTunes.

It's too bad I didn't chat with John Hockenberry after yesterday's skating.  Wow!  Russian skating is top notch, as proven in the team event.  Evgeni Plushenko is always a marvel to behold.  My younger daughter and I were rooting for Gracie Gold, and she delivered her personal best in a fantastic performance.  But, we could hardly keep still while 15-year-old Julia Lipnitskaia was on the ice.  Flexibility, grace, power, and full athletic prowess were on display as she nailed one jump after another and finished with a spectacular spin featuring three separate positions.  Angular momentum conservation was front and center throughout!

Physics of a Dreadful Super Bowl

The Super Bowl was a real clunker.  If you are a fan of the Seattle Seahawks, you loved watching your team beat the Denver Broncos, 43-8.  If you are a Denver fan, you probably turned the game off at some point in the 3rd quarter.  For the rest of us who just wanted to see a competitive game, the Super Bowl was a real letdown.

Given that I wasn't on the edge of my seat with the action, I had a chance to look at the game with my physics cap on.  Below are some physics thoughts I had while watching the game.

• The game started at 49 F, which was much warmer than the predicted weather when I contributed to a piece on a cold-weather Super Bowl.  Cold weather effects were thus minimal and not a factor in the Broncos looking like Elway's first three trips to the Super Bowl.

• 9:45 in the 1st quarter:  It was 2nd and 7 at the Denver 39.  Demaryius Thomas caught the ball at the Denver 40.  Kam Chandler at 232 lbs made a monster tackle.  Execution was perfect as Thomas was knocked back 5 yards after being hit just a bit above his center of mass.  Imagine hitting a one-ton car going nearly 3 mph.  That's what the hit felt like for Thomas.  That 3 mph doesn't sound like much until you realize you're hitting a one-ton car!  In other words, you're not going anywhere!

• 5:49 in the 1st quarter:  Russell Wilson faced 3rd and 4 at the Seattle 39.  He fired a pass to Doug Baldwin at the Denver 49.  The 10-yard pass took just 0.4 seconds.  Wilson fired the ball with an average speed of 51 mph, which is what was needed for the 1st down.  Wilson threw much harder, more accurately, and had a better spiral than the future Hall-of-Famer Peyton Manning.

• 4:24 in the 1st quarter:  at the Denver 43, Wilson faced a 3rd and 5.  He threw the ball from the Seattle 49 while Doug Baldwin was just crossing the Denver 38 (the 1st down line, by chance).  Baldwin caught the ball at the Denver 20, meaning Wilson timed the pass to hit Baldwin 18 yards down the field from where Baldwin was when the ball left Wilson's hand.  The timing was perfect!  Baldwin made it to the Denver 6.

• What I mentioned above is how Seattle dominated the 1st quarter.  Tackling was more fundamental, timing was better, and physics was utilized in ways that Denver could only dream of.

• 12:05 in the 2nd quarter:  Seattle had 2nd and goal.  It was 8-0 at the time, when Marshawn Lynch got the ball at the 4.  He was hit at the 1-yd line, but the hit was off-center at his center of mass, which meant he merely bounced off the hit and into the end zone.  Again, poor tackling from Denver.

• 11:24 in the 2nd quarter:  Denver was at their own 22.  Manning threw to his left to Demaryius Thomas.  The first hit was at the 24-yd line.  Byron Maxwell went below Thomas's center of mass, followed by help from K.J. Wright and Bobby Wagner.  The first hit stopped Thomas's linear momentum because the hit was just below the center of mass, thus creating just enough torque to arrest the motion.  Thomas was in trouble because as the help came, he had no forward momentum.  It was the perfect gang tackle.

• Percy Harvin went 87 yards to open the 2nd half.  Harvin averaged 15.5 mph during his run.  He was speeding in a school zone!  His top speed was nearly 20 mph as he left Denver defenders in his dust.  Seattle won all phases of the game, most especially special teams.

• 3:06 left in 3rd quarter:  Wilson hit Jermaine Kearse at the 17-yard line of Denver.  Three Denver defenders all hit Kearse at his center of mass, which meant Kearse just bounced off the tackles.  First, Danny Trevathan hit Kearse right in the center of mass, followed by Wesley Woodyard, and then Duke Ihenacho.  It was terrible tackling by Denver that led to Kearse scoring.  For some reason, Denver simply had poor fundamentals and poor physics in their tackling.

• 2:39 in the 3rd quarter:  1st and 20 for Denver at the Denver 10:  Manning hit Wes Welker, and then Seattle's K.J. Wright hit Welker just below Welker's center of mass, ending Welker's progress immediately.  It was a textbook tackle and a perfect example of Seattle utilizing the principles of physics much more than Denver.

Seattle obviously dominated Denver, but I sat there as a physicist and marveled at how well Seattle's execution of play after play was consistent with what physics predicts for great football.  The physicist in me liked what he saw even if the game failed to live up to the hype!

How will cold affect the Super Bowl?

There is much talk this year about the Super Bowl in a cold-weather city.  How will the cold affect play this Sunday at MetLife Stadium in East Rutherford, NJ?  Click here for a story I worked on with Kristian Dyer at Metro New York.

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!

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!

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.

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!

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.

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?

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!

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???

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!

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!

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.

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!