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.