30 January 2012

Tennis Balls and Kinetic Energy

In the first volume of The Feynman Lectures on Physics, Richard Feynman writes, "It is important to realize that in physics today, we no knowledge of what energy is."  Though that quote comes from a book published in 1963, we are in no better position today, nearly a half century later, of knowing what energy actually is.  We use energy concepts all the time to calculate all kinds of wonderful things about nature.  We have all sorts of conceptual ideas of how to understand the application of equations for energy, but, like Feynman wrote, we really don't know what energy is.

Just this morning, I derived the "work-energy theorem" in my Classical Mechanics course, a derivation that always leaves a chill on my spine.  In short, that theorem states that the net work done on an object equals the object's kinetic energy change.  The net work done is independent of the path taken to get from starting point to ending point, and the notion of kinetic energy, or energy of motion, allows for the use of scalars instead of pesky vectors like force and displacement.  The kinetic energy is ½mv2, where m is an object's mass and v is its speed measured in some reference frame.  The beauty of the derivation is that kinetic energy is not assumed at the start.  We simply evaluate the work integral for the net force and out pops this thing ½mv2 that must be evaluated at the starting and ending points.  Only then do we call that thing "kinetic energy."

While watching yesterday's Australian Open men's final, I saw serves reaching speeds around 110 mph (177 km/hr or 49 m/s).  Given that a tennis ball weighs about two ounces, its mass is therefore about 56.7 grams (or 3.9 millislug, if you really want to use those units!).  Using SI units, a served tennis ball's kinetic energy is thus around 68.5 joules (0.016 nutritional calories or 0.065 Btu or 50.6 ft-lbs).  Of course, the ball's speed goes down on its way to the other side of the court because of air resistance, but something like 70 joules is a reasonable kinetic energy for a professionally-served tennis ball.

Now, think about what speeds some other sports balls would need to have in order to have a kinetic energy of 68.5 joules.  To keep things simple, assume all balls, including our tennis ball, have no spin.  Including spin is not hard, but I'll save a discussion of rotational kinetic energy for another blog post.  A 5-ounce (142 grams) baseball needs to travel 69.6 mph (112 km/hr or 31.1 m/s), whereas a 440-gram (0.97 pounds) Jabulani football needs to travel 39.5 mph (63.5 km/hr or 17.7 m/s).  A teenage boy can throw a baseball 70 mph and a teenage girl can kick a Jabulani football 40 mph.

The lesson here is that some kinetic energies are easier to achieve than others.  Of course, if your technique is good enough to launch a tennis serve at professional speeds, you still have to be able to control it!

29 January 2012

Six Hours of GREAT Tennis!

Okay, my title exaggerates by seven minutes the length of the Australian Open men's final.  But, Novak Djokovic's victory over Rafael Nadal is something I'll never forget.  What an epic match!  Djokovic and Nadal probably showed us the limits of what human beings can do on a tennis court.  When Djokovic broke Nadal in the 11th game of the 5th set, one had to think that Nadal was finished.  Nadal managed a break point in the 12th game, but Djokovic was not to be denied.  It was painful to see one of those men lose; Nadal was only second by the slightest of margins.  I am in awe after seeing the longest Grand Slam final.  Serbia will be celebrating for sure!

I regret not seeing Victoria Azarenka's title win over Maria Sharapova.  Though not as competitive a final as the men's final, I congratulate Azarenka for taking over the #1 spot in women's tennis.  Belarus has quite a star!

Australia gave us an amazing fortnight of tennis.  Let's hope Paris will be just as thrilling!

Look for another tennis post tomorrow.  The energetics of the ball fascinate me.

16 January 2012

Rafael Nadal and Newton's Third Law

I caught a glimpse of Rafael Nadal's opening match against Alex Kuznetsov.  The world's #2-ranked men's tennis player easily dispatched Kuznetsov in straight sets, thus kicking off Nadal's efforts to win a second Australian Open.

Newton's laws have been on my mind of late, and watching a tennis ball in flight brought the third law to the front of my mind.  One must employ Newton's second law if one wishes to model the trajectory of a tennis ball in flight.  Instead of thinking about that, I thought of the remarkable subtleties in Newton's third law.  Two objects exert forces on each other of equal magnitudes and opposite directions.  Some refer to this idea as "action/reaction," but I can't stand those terms.  When I hear "action/reaction," I think that one force is the "action," and then the other force comes along a little later as the "reaction."  That's not what happens!  One object exerts a force on another object at exactly the same time as the second object exerts a force on the first object.  One force does not precede the other.

The third law also tells us that forces, much like the Sith in Star Wars, come only in pairs.  There is no such thing as an isolated force.  Because the idea of a force requires two objects, Newton's third law pairs never appear on the same object.  Think about Nadal's powerful serve.  After the ball leaves his racket, and before it reaches the court's surface on the other side of the net, there are two forces on the ball.  One comes from the air (drag, Magnus, and buoyant forces are all air forces); the other comes from the Earth (gravity).  The third-law pair to the former force is a force on the air from the ball; the third-law pair to the latter force is a force on the Earth from the ball.  Think about that.  The ball exerts a force on the Earth of exactly the same magnitude that the Earth exerts a force on the ball!  Further, the ball pulls the Earth up while the Earth pulls the ball down.  That's true whether the ball is in flight or in Nadal's pocket.  Many people new to physics often have trouble with this idea.  Take the tennis ball's mass to be 58 grams.  That's a tad more than 2 ounces or almost 0.57 newtons.

So, do you believe that a tennis ball pulls up on the Earth with 2 ounces or 0.57 newtons of force?  To believe it, one may need to think about Newton's second law.  Sure, the tennis ball and Earth exert equal and opposite forces on each other, but we see only the effect of the Earth's force on the ball.  We don't see the Earth move!  That's because the Earth has a mass that is 100 trillion trillion times that of a tennis ball.  Drop a tennis ball, and it accelerates to the ground at about 9.8 meters per second per second (that's about 22 mph each second).  That acceleration is due to the Earth pulling on the ball with 2 ounces or 0.57 newtons of force.  The Earth hardly notices that same magnitude of force on it from the ball.  It's upward acceleration is 100 trillion trillion times smaller than that of the tennis ball.  Drop a tennis ball from a height of about 1 meter (a little more than 3 feet).  It takes about 0.45 seconds for the ball to hit the ground.  In that time, the Earth moves only about one trillion trillionth of a centimeter, which is 16 orders of magnitude smaller than the width of a hydrogen atom!  Suffice to say, the Earth couldn't care less that a tennis ball is pulling on it with 2 ounces of force!

11 January 2012

Steven Gerrard and Newton's Second Law

Liverpool beat Manchester City by the score of 1-0.  The lone goal for the Reds came via penalty kick in the 13th minute by Steven Gerrard.  Liverpool is but a game away from Wembley Stadium!  I mentioned Gerrard in an article I was invited to write for Physics Today that came out during the 2010 World Cup.  Click here for that short, general audience article (click here for the same article in Japanese).  Gerrard got his penalty kick just past Manchester City goal-keeper Joe Hart; the ball sneaked into the lower left portion of the goal.

Newton's second law popped into my head when I saw Gerrard's kick.  An object's mass multiplied by its acceleration is equal to the net, external force acting on the object.  As an equation, we might write that as ma = F.  Note that I do not write F = ma, which I choose not to do for pedagogical reasons.  As simple as that equation appears to be, it is quite subtle to work with upon meeting it the first time.  I actually wrote a general audience paper on why I write Newton's second law equation backward from what is conventional.  Click here for that article.  When I teach that equation to my students, I want to them to be aware that there is no force ma acting on the object.  I've lost count of the number of free-body diagrams that I've seen with ma forces acting on objects!

All the forces acting on an object with mass m are added as vectors and put on the side of the equation where F sits.  To analyze the motion of an association football, take the football's mass to be m, and note that a is the acceleration of the ball's center of mass.  A study of the ball's motion about its center of mass requires Newton's second law for rotations, which I won't discuss right now.  What Aristotle did not understand, and what made Newton famous, is that once the football left Gerrard's boot, Gerrard's influence on the ball came to an end.  The air (drag, Magnus, and buoyant forces are portions of the air's influence on the ball), Earth (gravity), and ground (also Earth, but I'm thinking grass now) act on the ball as it rolls toward the goal.  Gerrard could do nothing to influence the ball's motion once the ball left his boot!

Newton's genius was recognizing that a (nonzero) net, external force is required to change an object's velocity.  Aristotelian thinking leads to the belief that a (nonzero) net, external force is required to maintain an object's velocity.  That is not true!  An object may have many external forces on it and still move at a constant velocity, as long as all those external forces add (as vectors!) to zero.  The beauty of Newton's second law equation is that there is an a on the ma side of the equation, not a v.

Note that once the ball left Gerrard's boot, it had to slow down.  There was no force in the direction of motion to speed it up.  There are, however, interesting things that happen with the drag force as the ball passes through what's called the "drag crisis," but I'll save that discussion for later!  For now, congratulations to Steven Gerrard and Liverpool.  Congratulations, too, to Isaac Newton for giving us a wonderful way to think about how the sports world works.  This year we celebrate 325 years since Newton's Philosophiae Naturalis Principia Mathematica (or Principia for short) was published.

10 January 2012

Alabama and Newton's First Law

Congratulations to the University of Alabama for winning the national championship in college football.  I sat in awe last night as I watched the most dominating defensive performance I've ever seen on a college football field. To do what Alabama did to an LSU team with such an impressive season is truly remarkable.  Alabama most certainly deserves its championship.

Watching Alabama's defenders reminded me of Newton's first law, which we apply quite well to the sporting world.  An object in motion with a constant velocity, a velocity that could have zero magnitude, will remain that way unless acted upon by a net external force.  A beautiful statement, right?!?  Any sporting event provides a setting to think of Newton's laws, but I was struck last night by how many times LSU was thwarted on offense.  Click here for the box score of last night's game.  LSU had 92 yards of total offense, 39 of which came on the ground.  So many times, LSU runners were smacked with the reality of Newton's first law.  Just as they reached a constant velocity, a large, external Alabama force met them in a direction opposite their velocity.  Sometimes, that large, external Alabama force reached the LSU runner before he even achieved top speed (because the runner is accelerating just before being hit in this case, Newton's first law is not applicable).

I could obviously use any play from last night's game to talk about all three of Newton's laws.  Instead, I chose to think fondly of the first law each time an LSU runner got smacked with a large, external Alabama force.  Newton's first law can be quite subtle when we first meet it.  I'm always amused when I watch a science fiction movie that has a ship in deep space with engines ablaze.  Hey, if the ship is going a tenth the speed of light, it'll keep doing so unless acted up by a net external force, right?  No need to waste fuel by accelerating closer and closer to the speed of light!  Each Alabama smack down on an LSU runner reminded me that Aristotle had it wrong, and Newton had it right.