Javascript required
Skip to content Skip to sidebar Skip to footer
Home / Physics Unit 3 and 4 Worked Solutions

Physics Unit 3 and 4 Worked Solutions

In "Bull Durham," the much-beloved movie about baseball, romance and metaphysics, one of the most memorable scenes involves the Durham Bulls manager (played by Trey Wilson) chewing the team out after another loss in an interminable losing streak. After heaving an armful of bats into the shower, the red-faced manager growls, "This is a simple game. You throw the ball, you hit the ball, you catch the ball. You got it?"

On the surface, this quip seems like the perfect encapsulation of America's pastime. But as anyone who's tried to throw a knuckleball, hit a curveball or catch a deep shot to center field will confess, baseball isn't as simple as it might seem. Crash Davis (Kevin Costner), the veteran catcher of the same movie, provides a glimpse of the game's subtle complexities when he tries to teach his hapless, but wildly talented, pitcher another lesson:

Do you know what the difference between hitting .250 and .300 is? It's 25 hits. Twenty-five hits at 500 at-bats is 50 points, OK? There's six months in a season -- that's about 25 weeks. That means if you get just one extra flare a week, just one, a gork, you get a ground ball, you get a ground ball with eyes, you get a dying quail, just one more dying quail a week, and you're in Yankee Stadium.

Crash is probably closer to the truth. Baseball is a game of numbers, and even though few ball players hold a degree in mathematics, every one is a practiced mathematician on the field. That's because baseball obeys the same physical principles that govern planetary motion, car crashes and airplane flight. Yes, the spirit of Isaac Newton is alive and well on ball diamonds all over the world. And although a home-run hitter or a Gold-Glove-winning third baseman might not think about Newton or be able to recite his three laws of motion, he understands these and other physical principles intuitively.

This article will explore some of those physical principles and attempt to reveal the science behind the intuition. And just to make things interesting, we'll take a cue from the Durham Bulls manager, breaking down the physics of throwing, hitting and catching, in that order. That puts us on the mound, 60.5 feet (18.4 meters) from home plate, to start things off.

Let's throw the first pitch.

Pitching: It Starts With the Legs

When people talk about pitching, they often discuss how fast the ball travels to the plate. But where does a high-powered fastball get its energy? As Hall of Fame right-hander Early Wynn once said, "A pitcher is only as good as his legs." That's because a 94-mph (151-kph) fastball begins its life below the pitcher's belt, gathering energy from the legs and unleashing it through the trunk of the body and then to the arm.

To generate all of that energy, a pitcher must be able to throw an energetic ball consistently and accurately without unnecessarily straining the body. Although form and technique can vary wildly, most successful pitchers incorporate seven basic components into their pitch delivery. Let's walk through them, assuming a right-handed pitcher throwing with no runners on base:

  • Starting position -- The pitcher stands tall, with his body facing the plate and his glove, concealing the ball and throwing hand, tucked just below the chin. Both feet rest on the rubber, about shoulder width apart.
  • The rocker -- Now the pitcher takes a small step back with his left foot. This rocks the body slightly back, although the pitcher's chin and glove remain centered over his right leg. This slight rocking motion unlocks a flow of momentum and energy that cascades through the pitch.
  • The pivot and balance -- The pitcher transfers weight again by making a small turn with his right foot to place it parallel with the rubber. This is the pivot, which turns the body so only the left hip and shoulder face the plate. Next, the pitcher moves into the balance position by lifting his left knee smoothly until it's level with his waist. Both hands should still be together and located close to the body, under the chin.
  • Hand separation -- The balance position stores the pitcher's energy, much like a coiled spring. To unleash that energy, the pitcher drives forward with his right leg and hip. At the same time, the hands first separate, then move down, back and up like a pendulum. The left foot, still off the ground, leads the body forward.
  • Power slot position -- The pitcher plants his left foot in front of his body while the right foot remains situated against the rubber. His throwing hand now sits level with his hat, fingers on top of the ball. His left arm is bent at the elbow, with the gloved thumb pointing to the ground.
  • The square position -- With his left foot firmly planted, the pitcher rotates his hip and trunk by turning on the ball of his right foot. This squares the pitcher's chest, shoulders and hips with the plate and starts to bring his throwing hand forward.
  • The delivery -- Finally, the pitcher drives his right leg forward, which rotates his hip around his front leg. This in turn rotates the torso and brings the arm forward. He makes the throw by whipping his right arm from 11 o'clock to 5 o'clock and then following through so that his right foot touches down with his body square to the plate, ready to field a ball batted directly back.

The ball now carries the energy generated in the pitcher's windup and delivery. How much? Physicists would use the following formula:

K = 0.5mv2

where K is kinetic energy, m is the mass of the baseball and v is its velocity. If the ball reaches a speed of 94 mph (42 meters per second), the kinetic energy would be (0.5)(0.149 kilograms)(42 meters per second)(42 meters per second), or 131 joules. A speeding bullet carries about 4,000 joules of energy, so a "rifle arm" is an overexaggeration, but it's still a lot of energy in the sports world (a well-hit serve in professional men's tennis would give a tennis ball about 128 joules of kinetic energy) [source: Impey].

Forces Acting on Pitches

We may play baseball on the moon someday, but it won't be very exciting, especially for pitchers. That's because pitchers rely on air resistance to throw breaking balls like curveballs and sliders and keep batters guessing. Understanding how this works requires a crash course in aerodynamics, a branch of physics that studies the properties of moving air, as well as how solid bodies interact with air as they move through it.

A baseball thrown by a pitcher pushes aside air molecules as it races toward the plate. As they encounter the leading edge of the ball, the molecules stream up and over and back together again to produce a wake behind the ball. For a slowly moving ball, the total pressure acting on the ball remains constant, but the "stickiness" of the air creates a frictionlike force known as viscous drag. As a ball moves increasingly faster, drag becomes more complicated. For such a ball, air doesn't actually reach the surface but forms a smooth, quiet boundary layer around the sphere. At speeds below 50 mph (80 kph), this mini whirlpool of circulating air molecules remains intact, and the air flow is smooth all the way around the ball. As the ball speeds up, however, frictional forces begin to peel away the layer, which creates an area of turbulence -- and lower pressure -- behind the ball. Higher pressure at the front of the ball gets the upper hand and exerts a second retarding force known as pressure drag.

Spin also affects air flow. A baseball leaving a pitcher's hand with significant spin will experience a force, known as the Magnus force, that acts at a right angle to the axis of spin. Consider a thrown ball spinning counterclockwise as viewed from above the mound. The air will flow faster around the third-base side of the ball than the first-base side. On the faster side, the boundary layer will peel away farther upstream, deflecting the trailing wake toward third base. The drag on the third-base side is greater than the drag on the first-base side, so the ball curves toward first base, at a right angle to the axis of spin.

The Magnus force explains how pitchers throw, as Crash Davis would say, "ungodly breaking stuff." For example, to throw a fastball, a pitcher places his middle and index fingers relatively close together and across the laces. When the ball comes out of the pitcher's hand, it has significant backspin, which means its spin axis is parallel to the field. The Magnus force, in this case, pushes up on the ball, acting against gravity.

Hitters often say such a pitch "hops" as it approaches the plate, but it's more likely that the Magnus force simply keeps it from dropping as quickly as it normally would. A good curveball, on the other hand, has more spin but less velocity. Not only that, if it's thrown well, it will have both topspin and sidespin, which places the axis of spin somewhere between horizontal and vertical. The Magnus force also acts on an angle, causing the ball to break down and to the side, sometimes as much as 15 inches (38 centimeters)[source: Nathan]!

Breaking Down the Swing, Millisecond by Millisecond

Facing a world-class pitcher might be one of the most intimidating prospects in all of sports. Consider the situation for a typical batter: He stands at home plate, awaiting the throw from the opposing pitcher, positioned just 60.5 feet (18.4 meters) away. A good fastball pitcher can hurl the ball at about 94 mph (151 kph). That means the ball covers the distance between the mound and home plate in 0.439 seconds, or 439 milliseconds.

To put that in context, remember that a voluntary blink of the eye requires 150 milliseconds. In a little less than three blinks, the batter must see the ball, judge its trajectory, decide whether to swing and then, potentially, swing. It's no wonder hitting safely just three out of 10 at-bats is considered good. Or, as Ted Williams once observed: "Baseball is the only field of endeavor where a man can succeed three times out of 10 and be considered a good performer."

Successful batters have two qualities you hear shouted every day at ball fields: "good eyes" and "quick hands." If we dissect a batter's reaction, millisecond by millisecond, this is what we get [source: Adair]:

  • The instant the ball leaves the pitcher's hand, light bounces from the leather surface and races to the batter's eye. There, it takes 75 milliseconds for the batter's eye to form a picture and send it to the brain: 25 milliseconds for the retina to process incoming data, 20 milliseconds for that data to travel to the brain and 30 milliseconds for the brain to construct an image out of the information. As the ball travels toward the plate, additional images arrive in the brain every 25 milliseconds, like frames in a motion picture.
  • The batter then has about 50 milliseconds for thinking. In this short time, his brain must analyze the pitch and select a suitable swing or decide that the pitch will land out of the strike zone.
  • Next, the batter's brain must send signals to the muscles of his legs, torso and arms. This manifests itself as electrochemical impulses originating in the cerebral cortex and then zipping along nerve fibers to the extremities. In all, it takes 25 milliseconds for the impulses to get from the brain to individual muscle fibers in the lower legs.
  • Finally, the muscles respond, but not instantly. A baseball swing requires a full 150 milliseconds to complete. This takes into account the time for muscles to contract, move their attached limbs and bring the bat around at a speed close to 80 mph (129 kph). That means a good batter is well into his swing when the ball is still about 18 feet (5 meters) from home plate. If he's 10 milliseconds too early or too late, he'll miss the ball completely.

That's all timing. What about the swing itself?

How to Swing a Bat (From a Physics Perspective)

The swing itself breaks down into two phases. At the beginning of the first phase, the batter stands with his feet spread to shoulder width, with the bat held nearly upright over the back shoulder (right shoulder for a right-handed hitter). As the ball approaches, the batter shifts his weight to his back foot and then pushes off, stepping forward with his opposite foot. This moves the batter forward, although the bat remains in its original position.

The second phase of the swing begins after the batter gets his front foot firmly planted. This foot then serves as a pivot point around which to rotate the body. Energy from the first step moves up through the legs and joins rotational energy created by muscles in the hips and torso, which pull the upper body around.

At first, the batter keeps his elbows bent, and the bat gains relatively little speed as it stays close to the body, moving in a small-radius arc. Over a few milliseconds, his arms straighten, and the bat levels out until it lies in a horizontal plane. Both the bat speed, and the radius of the arc it traces, increase dramatically through this part of the swing. When the bat finally makes contact with the ball, with both arms pulling at right angles to the motion of the handle, the barrel is moving between 70 and 80 mph (113 and 129 kph) and can generate 6,711 watts of power (9 horsepower) [source: Adair].

The Bat, How Sweet It Is

Yogi Berra once said, "I never blame myself when I'm not hitting. I just blame the bat, and if it keeps up, I change bats." In youth ball, changing bats usually means grabbing more aluminum. In the Majors, it's wood, or it's nothing. This may not seem significant, but bat construction has important implications for the game.

First, aluminum bats are hollow, while wooden bats are solid. That means you can make aluminum bats longer or fatter through the barrel without making them unnecessarily heavy. Not so with wooden bats, which get heavier as you make them longer or thicker. Another key difference is the location of a bat's center of mass, or CM. For aluminum bats, the CM is closer to the handle; for wooden bats, it's closer to the end. As a result, aluminum bats are much easier to swing, which means batters get them around faster.

Does this affect batted balls? As it turns out, increasing the mass of a bat increases batted ball speed, or BBS, but not nearly as much as increasing the velocity of the bat. According to some research, doubling the weight of a bat increases BBS by about 17 percent. Doubling the swing speed, however, leads to a 35 percent increase in BBS. In fact, for every 1 mile per hour you add to your swing speed, the ball travels an additional 8 feet [source: Coburn].

For these reasons, governing bodies at all levels of baseball closely regulate bat specifications. In Little League Baseball, the bat "shall not be more than thirty-three (33) inches in length nor more than two and one-quarter (2¼) inches in diameter. Non-wood bats shall be labeled with a BPF (bat performance factor) of 1.15 or less." BPF measures the increase in the liveliness of a ball hitting a bat compared to throwing a ball against a solid wall. So, if a ball has a 20 percent faster rebound coming off a bat, its BPF would be 1.20. Major League rules are more restrictive: "The bat shall be a smooth, round stick not more than 2¾ inches in diameter at the thickest part and not more than 42 inches in length. The bat shall be one piece of solid wood."

The material of a bat has no effect on its sweet spot, at least as it's defined in physical terms. Players know they've engaged the sweet spot when their hands feel no sting during the bat-ball collision. Or, as Ryan Zimmerman, the third baseman for the Washington Nationals, describes it: "Every ball I've hit that I haven't felt, I knew I hit well." Physicists now know that this sensation (or lack of sensation) is related to how a bat vibrates when it strikes a ball.

Yes, just like guitar strings, bats vibrate at multiple frequencies. When they vibrate at their fundamental, or lowest, frequency, they have a node -- a point at which the amplitude of vibration is zero -- about 6.5 inches (16.5 centimeters) from the barrel end. A second node, related to another frequency, appears 4.5 inches (11.4 centimeters) down the barrel. The space between these two nodes is the sweet spot, and hits involving the narrow zone produce less vibration and transfer more energy to the ball.

The Baseball, All Five Ounces of It

There have been a lot of rule changes to affect baseball over the years, but one of the most significant came in 1872. That's the year officials standardized the most essential element of the game -- the ball itself. Since then, every baseball has been exactly the same, weighing 5 ounces (142 grams) and measuring 9 inches (23 centimeters) in circumference.

It's not just the finished dimensions that are regulated, either. The manufacturing process must also follow strict guidelines. To build a baseball, you start with a nucleus of compressed cork wrapped first in a layer of black rubber and then a second layer of red rubber. Around this go three windings of wool and one of cotton. Rubber cement holds the yarn in place until a two-piece leather cover can be applied using exactly 216 red, raised cotton stitches.

There's more to this than simple trivia. The size, shape and construction of a baseball affect every aspect of the game. Can you imagine playing baseball with a tennis ball? Both balls are about the same size, but they differ in one important attribute -- tennis balls have significantly more bounce. In baseball, bounce relates to the "liveliness" of the ball. Before 1911, which is when cork-centered balls entered the game, hitting home runs was more difficult because balls didn't have as much bounce.

To understand the physics of a baseball bounce, it helps to analyze a dropped ball to see what it does. Over the years, several researchers have run this exact experiment, using high-speed cameras. This is what they've learned:

  1. When you drop a baseball, gravity pulls it toward the floor. All of the ball's potential energy, a function of its mass and height, gets converted into kinetic energy.
  2. When the ball strikes the floor, it stops for a split second. As it does, its kinetic energy becomes zero.
  3. The energy doesn't disappear. Instead, it gets used to change the shape of the ball. That's right, at the moment of impact, the ball compresses, transforming the ball from a perfect sphere into a slightly ovoid shape. At a molecular level, the long-chain polymers making up the ball's core stretch and squeeze.
  4. The stretching lasts only a fraction of a second. Then the polymers snap back to their original shape, and the ball begins moving up. Physicists would say that the ball's kinetic energy changed to elastic energy and then back to kinetic energy. Some energy, however, is lost to heat during the compression, so the ball doesn't return to its original height. In fact, a baseball will only bounce back to about 32 percent of its original height [source: Doherty].

Major League Baseball takes great pains to control the bounciness of its balls. After they're manufactured, balls travel to a testing facility where they're shot from a cannon at a velocity of 85 feet (26 meters) per second against a wood surface 8 feet (2.4 meters) away. Officials measure the rebound speed and then divide that value by the delivery speed. The resulting value must be 54.6 percent, plus or minus 3.2 percent. In other words, the rebound speed must be about 46 feet (14 meters) per second. Balls must also retain their shape after being subjected to a 65-pound (29-kilogram) force and distort less than 0.08 inches (0.2 centimeters) under compression [source: Nathan].

The Bat/Ball Collision: Part 1

On the last page, we discussed what happens to a baseball that's dropped straight down to a hard, flat surface. The same principles apply when a bat collides with a ball, but much larger forces and energies are involved. Remember, a pitched ball approaches 100 mph (161 kph), while a swung bat travels between 70 and 80 mph (113 and 129 kph). Bring the two together, and a tremendous force acts on the ball in one millisecond.

At the moment of impact, the ball compresses and comes to a momentary halt. The bat also compresses, though not as much. Then the ball reverses direction and leaves the bat at a certain velocity and at a certain angle, returning to its original shape as it does. The force acting on the ball and its compression depend, of course, on the intensity of the batter's swing. The stats below shows the peak force (in pounds) and compression for a ball hit with different types of swings [source: Adair]:

  • Bunt: peak force of 3,100 pounds and peak compression of 0.38 inches (0.97 centimeters)
  • Short fly ball: peak force of 7,000 pounds and peak compression of 0.54 inches (1.37 centimeters)
  • Home run: peak force of 11,000 pounds and peak compression of 0.69 inches (1.75 centimeters)

As we've already discussed, the compression of a baseball is an inefficient process. A large fraction of the ball's original energy dissipates as frictional heat caused by the squeezing, stretching and rubbing of the polymers that make up the ball's internal structure. Baseball geeks and science types describe this inefficiency using a value known as the coefficient of restitution (COR). For a perfectly elastic collision, one in which no energy is lost, the COR would equal 1. For a perfectly inelastic collision, the COR would equal zero (and the two colliding objects would stick together -- think Silly Putty smacking a surface).

To calculate a ball's COR, you take its velocity before it collides with a hard, immovable surface and divide it by its velocity after the collision. Recall that this is the test we described in the last section, which would look like this as an equation (note: for simplicity's sake, we'll omit the metric conversions here):

COR = velocity of ball shot from cannon/velocity of ball rebounding from wood surface =

Vi/vr = 46 feet per second/85 feet per second = 0.54

For collisions typical in baseball, this is the expected value for the ball's COR. As a measure of ball liveliness, the COR has important implications for the game. Increase the COR, and hit balls will travel much farther. For example, a ball with a COR of 0.6 would travel about 115 feet (35 meters) farther than a ball with a COR of 0.4 (assuming constant bat and ball velocity).

The Bat/Ball Collision: Part 2

Our discussion of the bat-ball collision has been very ball-centric so far. Another important measure is something known as collision efficiency, or q, a value related to a bat's ability to turn an incoming pitch into a solid hit. To calculate collision efficiency, physicists must consider both the ball's COR and the bat's COR, a value they call ball-bat coefficient of restitution, often abbreviated BBCOR. When they do, they get values for q in the range of 0.2 to 0.25. This in turn can be used to calculatebatted ball speed, or BBS, via the following equation:

BBS = (q)(pitch speed) + (1+q)(bat speed)

Plug in some typical numbers, and you get this:

BBS = (0.2)(94 miles per hour) + (1.2)(70 miles per hour) = 18.8 miles per hour + 84 miles per hour = 102.8 miles per hour

A "hotter" bat would have a higher q -- let's say 0.23 -- and would result in a higher BBS:

BBS = (0.23)(94 miles per hour) + (1.23)(70 miles per hour) = 21.62 miles per hour + 86.1 miles per hour = 107.72 miles per hour.

Batted ball speed has a direct impact on how far the ball travels. Increase BBS, and you put a fly ball closer to the fence. By relating BBS to known quantities, this equation reveals something important about the bat-ball collision: that bat speed matters more than pitch speed, but that the bat itself -- how it interacts with the ball -- plays a key role. Wooden and aluminum bats, for example, behave quite differently when they strike a ball. Both types of bats vibrate at the moment of impact, but wooden bats do so in one direction only -- along their length. These low-frequency bending vibrations dissipate much of the energy associated with the bat-ball collision, which means wooden bats don't return as much energy to the ball.

Aluminum bats vibrate in two directions -- along their length and radially as the metal shell squeezes in and then contracts out. This second class of vibrations occurs in a set of frequencies known as hoop modes. The fundamental frequency, or first hoop mode, acts like a spring during collision, compressing in and then expanding out and returning a large amount of energy to the ball. This is known as the "trampoline effect," and it's one reason why aluminum bats lead to higher batted ball speeds. It's also why aluminum bats ping when they strike the ball. The first hoop mode, with a frequency of 1,428 hertz, is associated with a tone that can be heard by the human ear [source: Russell].

The Fine Art and Physics of Fielding

The crack of the bat -- baseball's most iconic sound -- signals the end of a batter's stress and the beginning of a fielder's. The nature of that stress is quite different for a center fielder than it is for a third baseman. Let's start first with the physics of catching a fly ball by taking two common scenarios: a short pop-up that stays in the air about 5 seconds and comes down 250 feet (76 meters) from home plate, and a deep shot that stays in the air just over 4 seconds and lands 350 feet (107 meters) from home plate. If a center fielder positions himself 300 feet (91 meters) from home plate, he must move 50 feet (15 meters) in just a few seconds. Robert Adair, in his book "The Physics of Baseball," estimates that a typical outfielder can run about 30 feet (9 meters) per second, so getting to the ball would be relatively easy -- if he moved at top speed the instant the ball was hit.

Unfortunately, there's some processing time that must take place. For balls hit to his left or right, a center fielder will need 0.5 seconds to register the trajectory. That still gives him 3.5 to 4.5 seconds to get to the ball. But if the ball is hit directly at him, he will need two full seconds before he can sense whether he's dealing with a blooper or a monster blast to deep center [source: HowStuffWorks Videos]. That shaves his time to react considerably.

For these reasons, good outfielders analyze other factors before a pitch is even thrown. For example, they study their opponents' tendencies -- does a hitter like to hit to the opposite side of the field -- and cheat a few steps accordingly. They also watch the pitcher's delivery and the batter's swing for clues. If a ball breaks to the inside and "handcuffs" the hitter, the resulting swing will likely be far less powerful, allowing the outfielder to run in even before he gets a good look at the batted ball. Sound provides similar clues. A clunk indicates a collision outside the bat's sweet spot. A crisp crack is the telltale sound of a ball hitting squarely in the sweet spot. It only takes 0.3 seconds for either sound to reach a center fielder's ear, so listening to a hit can translate into one or two steps -- and the difference between a catch and a miss.

In the infield, fielders face different challenges, especially bad hops. First, consider the reality. If a batter hits a ball at 65 mph (105 kph), or 95 feet (29 meters) per second , toward third base, it will reach the fielder in about 950 milliseconds. The reaction time for a good infielder is 150 milliseconds, plus 50 milliseconds to move the glove [source: Adair]. That means he needs 200 milliseconds -- or 19 feet (5.8 meters) -- to make an adjustment. If a ball takes a bad bounce when it's closer than 19 feet, the fielder won't have enough time to correct his position and make a clean catch. If the bad hop occurs before then, say at 25 feet (7.6 meters), the fielder can make an easy adjustment.

Of course, he still has to make the throw to first. For a third baseman, this means slinging the ball 135 feet(41 meters) before the hitter covers 90 feet (27 meters). A fast runner will make it to first in about 3.0 to 3.5 seconds, so that's what the third baseman must beat. If it takes a second for the batted ball to reach the third baseman and another second for him to retrieve the ball from his mitt and throw, he only has 1.0 to 1.5 seconds to deliver the ball. How hard must he throw? Again, some simple math can answer the question.

If the third baseman makes a 90-mile-per-hour (132-feet-per-second) throw, the ball will arrive in the first baseman's mitt in 1.02 seconds, and the runner will likely be called safe. If he increases the velocity of his throw to 100 miles per hour (147 feet per second), the ball will smack the first baseman's mitt in 0.92 seconds, and the runner will be called out. Either way, it's a game of inches, which is why baseball attracts so many fans -- and encourages analysis by some of the world's most gifted scientists.

Author's Note

I played baseball for 10 or 11 years when I was a kid. I was a competent shortstop and an OK batter, and I loved playing the game even if I didn't understand all of the science. I wonder now, after writing this article, how the game would have been different if my coaches had taught me some of the physics behind pitching, hitting and fielding. Then again, what 10-year-old kid wants to know about the coefficient of restitution? Some concepts, even in baseball, are better left for an older, more mature player.

Related Articles

  • How Baseball Works
  • How Minor League Baseball Teams Work
  • In baseball, how does a pitcher throw a curveball?
  • When a baseball player hits a home run, how do they know how far the ball traveled?
  • How a Baseball Scorecard Works
  • 5 Must-learn Pitching Strategies

Sources

  • Adair, Robert K. "The Physics of Baseball," Third Edition. HarperCollins Publishers. 2002.
  • Fleitz, David. "Candy Cummings." Society for American Baseball Research. (July 30, 2012) http://sabr.org/bioproj/person/99fabe5f
  • Benson, Tom. "Aerodynamics of Baseball." NASA Glenn Research Center. May 9, 2012. (July 7, 2012) http://www.grc.nasa.gov/WWW/K-12/baseball/index.html
  • Coburn, Davin. "Baseball Physics: Anatomy of a Home Run." Popular Mechanics. Dec. 18, 2009. (July 7, 2012) http://www.popularmechanics.com/outdoors/sports/physics/4216783?click=main_sr
  • Cross, Rod. "Physics of Baseball & Softball." University of Sydney Physics http://www.physics.usyd.edu.au/~cross/baseball.html
  • Cross, Rod. "Mechanics of swinging a bat." American Journal of Physics. January 2009. (July 7, 2012) http://www.physics.usyd.edu.au/~cross/PUBLICATIONS/3.%20BatSweetSpot.pdf
  • Doherty, Paul. "That's the Way the Ball Bounces." Exploratorium Sport! Science. 1991. (July 7, 2012) http://www.exploratorium.edu/baseball/bouncing_balls.html
  • Glanz, James. "The Crack of the Bat: Acoustics Takes on the Sounds of Baseball." The New York Times. June 26, 2001. (July 7, 2012) http://webusers.npl.illinois.edu/~a-nathan/pob/glanz-nytimes.htm
  • HowStuffWorks Videos. "Physics of Baseball Fielding." (July 7, 2012) https://videos.howstuffworks.com/science-channel/4940-the-physics-of-baseball-fielding-video.htm
  • Impey, Chris. "Potential and Kinetic Energy." Teach Astronomy. (July 7, 2012) http://www.teachastronomy.com/astropedia/article/Potential-and-Kinetic-Energy
  • Little League Online. "Baseball Bat Rules for 2012." (July 7, 2012) http://www.littleleague.org/learn/equipment/baseballbatinfo/batrules.htm
  • MLB.com. "Official Rules." (July 7, 2012) http://mlb.mlb.com/mlb/official_info/official_rules/objectives_1.jsp
  • Nathan, Alan. "Evolution of the Baseball." The Physics of Baseball. University of Illinois. (July 7, 2012) http://webusers.npl.illinois.edu/~a-nathan/pob/evolution.html
  • Nathan, Alan. "Comparing the Performance of Baseball Bats." Baseball Analysts. Jan. 18, 2010. (July 7, 2012) http://baseballanalysts.com/archives/2010/01/comparing_the_p.php
  • Nathan, Alan. "The Physics of Baseball." University of Illinois. (July 7, 2012) http://webusers.npl.illinois.edu/~a-nathan/pob/index.html
  • Nathan, Allan, Daniel A. Russell and Lloyd V. Smith. "The Physics of the Trampoline Effect in Baseball and Softball Bats." Engineering of Sport 5. International Sports Engineering Association. 2004. (July 7, 2012) http://webusers.npl.illinois.edu/~a-nathan/pob/trampoline-v6.pdf
  • Nathan, Alan. "Wood versus Aluminum Bats." The Physics of Baseball. University of Illinois. April 16, 2007. (July 7, 2012) http://webusers.npl.illinois.edu/~a-nathan/pob/al-vs-wood.pdf
  • Nathan, Alan. "Working Knowledge: Baseball Pitches." Scientific American. September 1997.
  • Rist, Curtis. "The Physics of ... Baseballs." Discover Magazine. May 1, 2001. (July 7, 2012) http://discovermagazine.com/2001/may/featphysics/?searchterm=physics%20of%20baseball
  • Russell, D.A. "Hoop frequency as a predictor of performance for softball bats." Engineering of Sport 5. International Sports Engineering Association. 2004. (July 7, 2012) http://www.acs.psu.edu/drussell/bats/Papers/Russell_HoopFreq-ISEA2004.pdf
  • Russell, Daniel A. "Physics and Acoustics of Baseball and Softball Bats." Penn State University. (July 7, 2012) http://www.acs.psu.edu/drussell/bats.html
  • Solesky, Len and James T. Cain. "The Physics of Pitching." MVP Books. 2011.
  • St. John, Allen. "Should Metal Baseball Bats Be Banned?" Popular Mechanics. May 18, 2010. (July 7, 2012) http://www.popularmechanics.com/outdoors/sports/baseball/minor-league-metal-bat-ban

Physics Unit 3 and 4 Worked Solutions

Source: https://entertainment.howstuffworks.com/physics-of-baseball.htm/printable