Seeing a banked three-point shot from each of my alma maters last night got me thinking about bouncing basketballs. Hold a basketball at a certain height and drop it onto the floor. You will notice that it does not return to the same height from which it was dropped. The ball in fact returns to a lower height because of energy lost during the collision with the floor. There is a little energy lost to air resistance as the ball moves through air, but that loss is small compared to the energy lost during the collision. It is actually easy to tell that energy is lost, even with your eyes closed! The fact that you hear the ball bounce off the floor means that your ear picks up a sound wave that carries energy. Energy is also lost to heat as the basketball's surface rubs against the floor. The reason the ball rubs against the floor is because the ball deforms during the collision, meaning its surface spreads slightly against the floor. Using more technical language, the ball scrunches on the floor.
A parameter we use to tell us something about energy loss is the coefficient of restitution or COR, which is the ratio of the speed just after the bounce to the speed just before the bounce (this assumes the ground does not move). Because those speeds are sometimes hard to measure, we use heights. A simple way to calculate COR for the dropped basketball experiment that I mentioned in the previous paragraph is COR =( hf / hi )1/2, where hi is the initial height off the floor and hf is the rebound height after the bounce. According the NCAA rules (click here for the rule book and go to page 25), a ball dropped from a height of 6 feet (1.83 m) must rebound to a height of "not less than 49 inches" (1.24 m) and "not more than 54 inches" (1.37 m) for a college men's ball. A college women's ball must rebound between 51 inches (1.30 m) and 56 inches (1.42 m). That means that for a college men's ball, 0.825 < COR < 0.866, and for a college women's ball, 0.843 < COR < 0.882. A referee can check a ball before a game by dropping it from the height of his or her head and seeing if the ball bounces to a height roughly 2/3 to 3/4 of the referee's height, which is roughly the height of the referee's mid torso. A ball that does not satisfy the referee will need to have its air pressure changed if there is nothing wrong with the ball itself.
The amount of energy the ball keeps after the collision scales with the square of COR. A college men's ball thus retains between 68% and 75% of its energy after colliding with the floor. For the college women's ball, between 71% and 78% of the ball's energy is retained. A guard dribbling the ball down the floor puts energy into the ball by doing work on it, namely by exerting a downward force on the ball while displacing the ball downward. One must keep pushing the ball downward if one wants it to return to the same height each time because a ball with an initial downward velocity from a height of a player's hip can return to the player's hip height, whereas a ball dropped (i.e. zero initial velocity) from the player's hip height cannot.
Note that COR depends on the properties of the two surfaces involved in the collision. One cannot simply quote a COR for a given sports ball without also specifying the surface the ball hits. For the COR calculations I did here, I used the specifications in the rule book, which describe the ball colliding with the "playing surface." For a ball bouncing off glass, like in the banked three-point shots I saw last night, COR is likely to be different from the value measured by dropping a ball on the playing surface.
A bank shot is even more interesting than a ball dropped onto the floor because the ball banking off the glass not only feels a force from the glass that is perpendicular to the glass surface, the ball feels a frictional force parallel to the glass surface. That frictional force creates a torque on the ball, which changes the ball's rotation. That means that the rebound angle the ball makes with the glass is not the same as the angle the ball made with the glass just before the ball hit the backboard. Basketball players know this -- just watch where they bank the ball off the glass on layups.
As always, enjoy the wonderment of a sporting moment first, and then think about physics later. After both my schools won last night, I was too happy to write about physics!
Okay, so I've got a silly title for this blog post, and there won't be much sports science. Because I don't have any favorite professional teams, my rooting interests revolve around my two alma maters. It's not every day that I get to see my two schools play back-to-back on national television. ESPN showed my 20th-ranked Hoosiers hosting 5th-ranked Michigan State at 7:00 pm, followed by my Commodores hosting 13th-ranked Florida at 9:00 pm. Both my schools were underdogs, and both earned a double-digit victory.
Indiana, my graduate school, has beaten the #1-, #2-, and #5-ranked teams this season. We had Michigan State's number tonight (click here for the box score), leading by 14 at the half and winning by 15. Draymond Green for the Spartans was fantastic, scoring 29 of Michigan State's 55 points. He should easily be the Player of the Year in the Big Ten.
Vanderbilt, my undergraduate school, joined Indiana as a school with 10 conference wins. We shot well to knock off the highly-ranked Gators (click here for the box score) by 10 points.
My schools gave me four consecutive hours of wonderful basketball viewing tonight. I even got to see a player from each of my schools hit a three-point bank shot. A fun night indeed!
If you missed Cristiano Ronaldo's backheel goal to beat Rayo Vallecano today, find it on YouTube. Real Madrid's superstar's goal in the 54th minute provided the only scoring in the match. Almost 11 m (12 yards) from the goal, and nearly aligned with the right goal post, Ronaldo backheeled the ball, which sent it rolling toward the left portion of the goal. The ball traveled about 12.3 m (13.5 yards) from Ronaldo's heel to the goal line. It was an amazing demonstration of athletic ability and awareness on the pitch.
After picking up my jaw upon seeing Ronaldo's great goal, the first bit of physics that entered my mind was that of impulse. In physics we define impulse in one of two ways. It's the change in linear momentum (mass times velocity), which is the same as the net force times the collision time. This all comes from Newton's second law, but the details are not so important right now. We make use of this idea a lot in everyday life. Usually, the change in linear momentum is something we cannot control. Imagine driving in a car and having the misfortune to slam into a tree. Multiply your mass by the change in your velocity and that's your change in linear momentum. You can't change that because your car's speed goes from what it was before the collision to zero after the collision. Fortunately, there is another way to write impulse, as I noted earlier, and that is force times collision time (there is an integral here, but just think average force when I write force). If you can do something to extend the collision time, you can reduce the force needed to stop you while you are in contact with the tree. You already know I'm referring to an air bag. If it can increase the collision time by, say, a factor of ten, then the force on you goes down by a factor of ten. It's better to hit the air bag than to hit the windshield, steering wheel, or dashboard.
You can think of many other examples. If you jump from some elevation, you bend your knees upon hitting the ground so as to extend the collision time with the ground. Pole vaulters prefer landing pads to the ground. Long jumpers like the sand pit instead of hard ground. You wear a padded glove while playing baseball, again so as to extend the collision time when you catch a baseball. Padding in American football and boxing gloves are yet more examples of ways to extend collision times.
The concept of impulse also helps us get some idea of the size of the average force when a collision takes place. The collision time between Ronaldo's boot and the football is nearly 0.010 s. I estimate that the ball left his boot at a speed around 13 m/s (30 mph). It was rolling at about 1 m/s (2.2 mph) in the opposite direction just before Ronaldo kicked it. The magnitude of the balls' velocity change was therefore about 14 m/s (31 mph). With a 440-gram (0.97 pounds) football and a collision time of 0.010 s, the average force on the ball from Ronaldo's boot was about 616 N (138 pounds). That might seem like a large force, but that average force lasts only ten milliseconds.
Think about Newton's laws and note that the Second Law tells us that the ball was slowing down the entire time after it left Ronaldo's boot. The Third Law tells us that Ronaldo's boot felt the same force that the ball felt during the collision. A little padding in the back of the boot helps extend the collision time between Ronaldo's heel and his boot. Football players are more than comfortable kicking the ball a lot harder than Ronaldo did. Of course, Ronaldo most certainly had no complaining from his heel after such a remarkable goal!
Two nights ago, Jeremy Lin of the New York Knicks hit a three-point shot to beat the Toronto Raptors. Tied at 87, Lin was just behind the three-point arc and slightly right of center when he let go of the game-winner. The ball fell through the net with just a half second remaining on the clock. Click here for the story and video. Lin has been in the news of late because of his great play, but also because of his great story. It's not all the time that we see an undrafted player from Harvard making the NBA highlight reel!
I had fun analyzing Lin's shot. After several timings, I estimate the ball's time of flight to be 1.32 s. He looked to be just under 24 feet (7.3 m) from the basket, and let go of the ball at the top of his jump from a height above the floor of about 9.8 feet (3.0 m). The basket sits 10 feet (3.0 m) off the floor.
To model the flight of the basketball, I included four forces. The ball's weight of 22 oz (0.62-kg mass) is a downward force. Because the ball displaces a volume of air equal to its own volume, the ball feels an upward buoyant force of about 0.31 oz (0.085 N), which is only 1.40% of the ball's weight. In a direction opposite the ball's velocity is the drag force due to air resistance; the size of the drag force depends on the ball's speed. The fourth force on the ball is the Magnus force, which is the same force that is responsible for curve balls in baseball and banana kicks in soccer. Lin let go of the ball with backspin, so the Magnus force, which depends on the ball's speed and spin rate, has a component that is upward. I estimate nearly three turns of the ball during its flight.
Using a computer to solve the ball's equation of motion that comes from Newton's second law, I get the trajectory in the image you see below (click on the graph for a larger image).
The ball left Lin's hand with a speed of nearly 19.9 mph (8.89 m/s) at 46.1° above the horizontal. The ball's speed dropped to 17.5 mph (7.83 m/s) by the time it went through the basket. Note that the ball reaches a maximum height of about 16.6 feet (5.05 m) above the court.
Below is a graph of the drag and Magnus forces on the ball as functions of time (click on the graph for a larger image).
Note that the forces in the above plot are at their minimum values when the ball is at maximum height, which is where the ball's speed is at its minimum value. When Lin released the ball, the forces in the graph are at their maximum values because the ball's speed is greatest then. The maximum drag force is 3.88 oz (1.08 N), which is 17.6% of the ball's weight. The maximum Magus force is 1.30 oz (0.361 N), which is 5.90% of the ball's weight. Note that the buoyant force, which I noted is 1.40% of the ball's weight, is nearly a quarter of the maximum Magnus force.
I hope we'll see more great shots from Jeremy Lin!
As I lick my wounds from seeing my beloved Vanderbilt Commodores fall to #1-ranked Kentucky last night, I reflect on the fact that on this day 203 years ago, Charles Darwin was born. Think about how much more human beings know about how our species evolved compared to what we knew two centuries ago. If you have only a superficial understanding of Darwin's contributions, please allow yourself to do a little reading. If reading a book published in 1859 (On the Origin of Species) is not your cup of tea, try, for example, The Greatest Show on Earth: The Evidence for Evolution by Richard Dawkins. Read about the 1860 evolution debates at Oxford, which featured Thomas Huxley and Bishop Wilberforce (among others) going toe to toe.
Darwin's work remains one of the greatest contributions to not only science, but to humanity. Evolution is not something one must "believe" or "take on faith." The evidence for evolution is immense, and only after the countless failed efforts to falsify Darwin's theories did the scientific community begin to accept them as real descriptions of the natural world. That is what we do in science. We make claims based on data and evidence, and perhaps extrapolate those claims to more far-reaching theories. Those theories are then put to the test over and over again by researchers all over the world. A good scientific claim must be falsifiable, meaning that if evidence is found to refute that claim, and the scientific community forms a consensus that the evidence found does indeed refute the claim, the claim is tossed out. We may not like being wrong, but we in science are not afraid of being wrong. That is how we learn!
I find it fascinating that Abraham Lincoln was born on the exact same day as Darwin. Deserving of the moniker, "The Great Emancipator," one could certainly argue that Lincoln was our greatest president. Some have suggested that Lincoln was actually not the greatest emancipator born on 12 February 1809! When one considers what Darwin gave science and humanity the world over, such a suggestion is not too far-fetched.
On a personal note, my paternal grandfather was born on 12 February 1922, which was 90 years ago today. He died in 1994, just five days after I turned 24. He always seemed tickled that he shared Lincoln's birthday. Because my scientific career was only just starting near the end of his life, and because my understanding and appreciation of Darwin's work came only in my mid 20s, I never talked to him about the fact that he also shared Darwin's birthday. I'm not sure if he knew that he did.
Tonight, I'll raise a pint to Darwin, Lincoln, and my grandfather. I learned a great deal from all three of them.
I got a comment from someone asking about Eli Manning's Hail Mary at the end of the first half of the Giants' divisional playoff game against the Packers on 15 January 2012. I had not modeled that pass, so I thank the person who left the comment for the suggestion.
Manning let fly his pass from the right hash mark at the Packers' 43-yard line. Hakeem Nicks caught the ball about 5 yards deep in the end zone. Nicks appears to have caught the ball about 20 yards to the left of the line of right hash marks. In other words, the ball went 48 yards straight and 20 yards left, leading to a horizontal range of 52 yards (Pythagorean Theorem!). After five timings of the flight time, I got a time of flight of 3.048 s.
Throwing the aforementioned numbers into my computer, I got a launch angle of 42.5 degrees and a launch speed of about 51.3 mph (82.6 km/hr). The maximum height reached above Lambeau Field was around 14.5 yards (13.3 m).
Manning did not have to throw the ball as far as Brady did at the end of the Super Bowl. Manning's pass was thrown about 88% of the speed of Brady's and at a slightly smaller angle. Brady's pass went about 5 yards higher, too. Credit Eli Manning for a great pass, but give equal credit to Hakeem Nicks for making a phenomenal catch. The Giants went into the locker room up 20-10 instead of 13-10. The play to end the half was a game-changer for sure!
After studying Brady's final pass, I analyzed two more plays that helped decide Super Bowl XLVI. Click here for the link to the article in YAHOO! SPORTS by Kristian Dyer.
I was one of the estimated 111 million people to watch the Super Bowl last night. That's a lot of people watching a football game, though still an order of magnitude less than the number of people that watch the FIFA World Cup every four years. Still, the Super Bowl is the biggest game of the year in the US. It was a great game that came down to a Hail Mary from future Hall-of-Fame New England quarterback Tom Brady.
Like everyone else, I held my breath while the ball was in the air. I later analyzed the throw because I was curious how well Brady had thrown the pass. Watching the replay over and over, I averaged five timings of the ball's flight time and got 3.474 seconds. Brady appeared to let go of the ball at the New England 42-yard line; the ball was first touched about 6 yards deep in the end zone. That means that the horizontal range of the ball was about 64 yards (58.5 meters). Solving the equation of motion from Newton's second law, which includes air resistance, I found that Brady released the ball with a speed of 58.4 mph (94.0 km/hr) at an angle of about 45.3 degrees from the horizontal. The ball reached a maximum height of roughly 19.5 yards (17.8 meters) above the turf.
It was a great pass, and it needed a lot of luck to be completed. After being tipped, New England tight end Rob Gronkowski dove for the ball and looked to have a change to catch it. But, alas, even at 6' 6" (1.98 meters) tall, Gronkowski was too late getting to the ball. Once the ball was tipped, it began accelerating to the turf at 32 feet per second per second (9.8 meters per second per second), which is about 22 mph per second. Gronkowski simply had too much distance to cover while the ball was making its way to the turf and giving the Giants their fourth Super Bowl win.
See Chapter 3 of my book, which focuses on Doug Flutie's famous Boston College pass to beat Miami in 1984, for more details on modeling the flight of a Hail Mary pass in football.
