I'm sitting on my warm, snuggly couch watching the Cleveland Indians and the Chicago Cubs in Game 2 of the World Series. The players, coaches, and fans are shivering on my television screen. Today, we are going to discuss the physics of home runs; unfortunately, the cold weather makes it unlikely anyone will hit a dinger in tonight's game. This phenomenon is rooted in fluid dynamics. Cold air has a higher density which results in increased drag on the ball. However, that discussion is outside the scope of this article.
We are going to investigate how launch angle impacts whether you fly-out or jog around the bases. Recall that the trajectory of a falling object is parabolic; meaning, it has an x- and a y-component. Knowing that distance travelled by a falling object is related to the initial velocity (v), we can mathematically express each component of the velocity utilizing concepts from trigonometry:
We get the following expressions after plugging the expressions above into the formulas for a zero launch angle:
Now we can really start digging into our problem. Let's make the following assumptions:
You may notice we don't have a time component, which is a problem since we have 3 unknowns and two equations. However, we can overcome this by reframing our question: How far has the ball travelled when it reaches a height of 3 m? We ask this question because we want to find the ideal launch angle to hit a home run. So the mechanics look like this:
Therefore, under the assumed conditions a launch angle of 45 degrees would result in a home run! Let's look at a range of angles to find the "sweet spot:"
It appears when the ball is traveling at 35 m/s the ideal launch angle is between 45 and 50 degrees. The next question is: How would this change with varying initial velocities?
As expected, you have greater flexibility of launch angle with increasing velocity. So if you want to crush the ball, you better start pumping iron!
This week we find ourselves in the middle of the Major League Baseball (MLB) Post Season. In the flurry of excitement we hear commentators spouting off different stats. Since everyone "digs the long ball" you better believe you are hearing exit velocity and pitch speed stats.
The exit velocity is the speed of the ball coming off the bat and - quite obviously - the pitch speed is the speed of the incoming throw from the pitcher. These stats are important elements of predicting the characteristic of the hit ball, particularly the ball's distance travelled. However, there has been much debate on the importance of pitch speed on the ultimate exit velocity. Some argue the faster the ball comes in the faster the ball will go out. However, this is not the case. To understand the impact of pitch velocity we need to look at the momentum of the ball.
In this system we consider the forces acting on the ball. Consider the set-up below:
The impulse (I) is the average force (Fav) delivered to the ball over time (delta t). This is also considered the change in momentum (delta p). The change in momentum can also be expressed as the initial momentum subtracted from the resulting final momentum. When we combine these two equations with the formula for momentum (mass multiplied by velocity) we see a relationship between the force delivered by the batter (Fav) and the speed of the pitch (vpitch). Since we are primarily interested in how these pieces interact to influence the exit velocity, let's isolate the exit velocity:
You may be thinking to yourself, "Whoa, the pitch velocity is added to the force applied by the batter. So faster pitches should lead to faster exit velocities." However, I would remind you to remember our coordinate system. If our pitch velocity is 90mph, we would write vpitch = - 90mph since it is traveling in the negative x direction. Therefore, pitch velocity decreases the ultimate exit velocity.
To give you an example, let's assume two pitches are thrown and the batter applies the same force to each pitch. How would the exit velocity differ? (We will assume the Fav term will equal 200mph for the ease of the example.)
This phenomenon can be observed in the MLB stats. Visit Statcast to see that some of this season's hardest hit balls were on relatively slow pitches. Another theory MLB player's have is that since the ball is traveling slower the batter has a better chance at hitting the ball square. There is some truth to this in the likelihood of turning a pitch into a "dinger." However, the physics shows us that if both balls are hit in the same manner that the ball hit off the slower pitch will go faster.
Enjoy the rest of the Post Season with this tidbit of information. Ask your friends watching the game what they think would result in a harder hit ball, then impress them with your insight.
Go Cubs! :P
Taking a test can be exhausting. It is not just the written portion that is exhausting, but studying for it as well. Many people think that the key to getting a good test grade is your ability to remember all of the information taught in class -- but that's only partially true. Every single piece of information presented in class was done in such a way that you would understand a few core concepts. Therefore, the key to studying for a test is to identify those core concepts and ensure you know how to apply them.
Identifying the Core Concepts
No test can cover everything discussed during class. Think about it. One chapter in a textbook may have 30+ pages choke-full of facts, figures, and definitions. On the other hand, one test is roughly 2+ pages of questions and blank space. How do you know what will be on those few pieces of paper?
Start by creating an outline. Read the objectives of the chapter in the textbook. Skim through the chapter and your notes from class. Ask yourself these questions:
Review your homework once you have created an initial outline. However, don't begin by re-doing the problem! Instead, look at the problem and ask yourself what that question was trying to teach you. It should teach you some element from your outline.
By the end of this exercise you should have something that looks like this:
What You Know Vs. What You Don't
There is no use spending hours pouring over information you already understand. The time spent studying is inversely proportional to how well you know it:
Compare your outline and homework again. Your homework is an indicator of how you will perform on the test. Ask yourself:
So your outline should start looking like this:
Pink is the "danger zone." You are at a high probability of getting any question containing this concept wrong. Yellow is "proceed with caution." Meaning, you may be able to answer the question but you do not fully understand why. You may do well on the upcoming test, but classes build on the previous chapters. Therefore, you will likely suffer on the next exam if you don't shore that concept up now.
Now that you have a plan you can start studying! Re-reading the same information over and over isn't the best approach. Instead, pretend you are a teacher attempting to present this information to the class. How would you make sense of it? Re-write your notes. Once I finished studying, I had my own study packet that was a hybrid of the teacher's notes, my observations, and information from the textbook.
Another study technique is to restructure the information. For instance, in biology you may want to organize the animal kingdoms into a flow chart. Pick a different color for each group. Tell yourself why you are picking that color, "Plantae is green because it is the plant group; plants are typically green." Even if your teacher already gave you a flow chart, draw it yourself from scratch. Reason through why each group is there and what it contains. It forces you to think more deeply than just looking at words on a page.
Another study approach is to think of examples. One friend told me how she explained the concept of a mole from chemistry as "A mole is to atoms like a dozen is to eggs." She equated the scientific concept to something we are familiar with in our everyday lives. Another friend of mine felt he had a good handle on a concept if he could come up with a joke that made the professor laugh.
Whatever it is for you, find a way to make the material engaging.
Work with Friends
In the beginning it can be helpful to study with someone that has developed great study habits. They can share what works for them and give you new ways to look at material. Learning does not happen in a bubble.
It takes time to develop study habits. You may try everything outlined above and realize you need to tweak a few things to make it effective for you. That's fine. We all synthesize information differently because none of us have the exact same frame of reference.
We would never expect someone to become a professional athlete the minute they pick up the ball. It takes years of dedicated effort to learn the necessary skills. This holds true for academics as well. Keep practicing and don't give up!