
English: 
Babe Ruth,  a legend of Major League Baseball
He led the league in home runs during a season 12 times
and made a record of 60 home runs in 1927 season
as well as 714 home runs of his life time record
In the dead-ball era
Ruth's home run and slugging percentage dominated the league
He creates an unchallengeable position in the heart of baseball fans
and was respectfully named "Baseball Demigod"
Roger Maris, an outfielder of New York Yankee
who challenged Ruth's record after 34 years-
- 60 home runs in a season

Chinese: 
貝比．魯斯  美國職棒的傳奇人物
曾經奪得12次全壘打王的頭銜
並且在1927年創下單季60支全壘打的紀錄
在其職業生涯總共累積了714支全壘打
在那個被稱為「死球時代」的時期
貝比．魯斯的長打能力遠遠的超越同時代的打擊者
於是  他在球迷的心中建立了不可挑戰的神格地位
被尊稱為「棒球之神」
羅傑．馬里斯  洋基隊的外野手
在30多年後挑戰貝比．魯斯的紀錄
單季60支全壘打

English: 
With his home run number getting close to 60
however, there was no joy for fans
because there can be only one hero in their heart
Roger, up here
Hey it's the voice from above
Hey Maris up here
it's the Babe
Hey you wnat my record? you want my record?
Catch this you ape piece of shit
Stop
In the game 154 in 1961
Maris hit number 59 in the 3rd inning
Two outs, in the top of 9th
and it's the last chance for Maris
Orioles was behind but made an unusual substitution
The closer Hoyt Wilhelm was called on to stop him tying Ruth's record
I hate that knuckleball
In this weather it's gonna be dancin' all over the place

Chinese: 
然而  眼看著馬里斯的全壘打數字步步逼近
球迷的心中並沒有喜悅
因為英雄只能有一個
嗨  羅傑  看上面
這是來自天堂的聲音  哈哈哈...
嘿  馬里斯  看上面
貝比．魯斯在此
你想破我的紀錄嗎?  想破嗎?
垃圾來了給我接好  猿猴
暫停
1961年  洋基第154場比賽
馬里斯在第三局擊出了第59號全壘打
九局上半  兩出局
馬里斯最後一次打擊機會
為了防止他追平紀錄
金鶯隊在落後的情況下做出不尋常的調度
王牌投手  霍依．威漢登板救援
我痛恨蝴蝶球
今天的風會讓蝴蝶球亂飄

English: 
Even if he gets wood on it it's not goin' anywhere
Bushlinger
Wilhelm takes the sign
but we all know what's coming
Does Roger Maris have one last home rum in him
he fouled it back
Roger had a good cut on that ball, but it's dancin' all over the place
Wilhelm's got it
Out
The knuckleball dominated the hitter and Maris failed to tie the record on that day

Chinese: 
就算能碰到球  球也飛不到哪去
小人
威漢收到暗號
但大家都知道會配什麼球
羅傑能從他手中擊出最後一支全壘打嗎
打成後方界外
那一球的揮擊角度很好 問題是球四處亂竄
威漢接到球
出局
受到蝴蝶球的壓制  馬里斯只能鎩羽而歸

Chinese: 
在棒球的世界中  蝴蝶球是一種極為特殊的球路
它能夠忽左忽右  或者突然的下墜
以幾乎不太規則的方式運動
也就是這樣能夠愚弄打擊者
接下來  我們將從空氣動力學的角度出發
了解蝴蝶球的原理
並且進一步的  介紹並分析其他各種變化球球路
講到流體力學  一般人最熟知的就是柏努力定律
但是在柏努力定律之外
還有其他重要的流體理論需要了解與認識
以便具備足夠的基礎知識
我們先來談邊界層理論
當流體流過障礙物時  會與障礙物發生交互作用
我們先來研究最簡單的情況
當流體以速度V流過一個平板
流體會跟物體表面發生摩擦

English: 
Knuckleball is a very special pitch in baseball games
It may suddenly change in direction during its flight
and dance all over the place with erratic movement
so that it can fool the batter
In the following we will investigate knuckleball
base on aerodynamics
Furthermore, we will then introduce various pitches of baseball
When it comes to aerodynamics, most people are familiar with Bernoulli's principle
But, to establish enough background knowledge
we need to learn more about the fluid theories
besides the Bernoulli's principle
First let's talk about boundary layer theory
When a fluid flows over a obstacle, they would interact with each other
Let's look into a simplest case
When a fluid passing through a flat plate with velocity V
it causes friction on the surface

Chinese: 
因為流體一般都是有黏滯性的
所以越往平板表面靠近  流速會越來越慢
直到趨近於零
這個流速逐步遞減的範圍
稱之為「邊界層」
在這個薄層內  黏滯力主導物理現象
有黏滯力的區域就不適用柏努力定律
然而  在邊界層外面
黏滯力沒有發揮什麼作用
流體的行為接近於無黏滯性的理想流體
所以可以使用柏努力定律加以描述
我們稍後會再回來談柏努力定律
現在先讓我們來關注邊界層內的現象
當流速V不大的情況下
邊界層內的氣流是穩定的
流體的流線就像千層派一樣
這樣的邊界層稱之為「層流邊界層」
障礙物的幾何形狀會影響邊界層
例如說從平板變成球體的時候
會有新的物理現象發生

English: 
In general, fluids are viscous
so the flow speeds decrease gradually when it get close to the surface
and end up at zero
The region where flow speeds gradually decrease
is called the "Boundary layer"
Viscosity dominate physical phenomenon in this thin layer
and the Bernoulli's principle does not apply in this region
However, viscosity has no effect on the region outside the boundary layer
where the fluid behaves like a inviscid ideal fluid
such that the Bernoulli's principle applies
We will back to Bernoulli's principle soon
Now pay attention to the phenomenon inside the boundary layer
If the flow velocity V is not large
The airflow inside the boundary layer is stable
and the streamlines look like a Mille Crepe
This is named a "Laminar boundary layer"
The geometric shape of obstacles have effects on boundary layer
For example, when substitute the plate by a sphere
a new physical phenomenon may occur

Chinese: 
我們把球體周圍的流線畫出來
綠色的部位代表邊界層
它是重要的區域
我們先來探討邊界層外面的區域
那是可以運用柏努力定律的地方
觀察一小塊體積單元的運動
因為棒球運動遠低於音速
不必考慮壓縮效應
這一塊小體積在運動過程中保持體積不變
而且可以把它想像成是在口徑不斷變化的管子中移動
當它靠近A點時
速度逐漸變慢
A點又叫做「停滯點」
流體會從停滯點開始往兩側分離
當它離開A點後速度逐漸增加
到達B點時速度最高
然後通過B點之後又逐漸減速
到達C點時流速又降到最低
柏努力定律是說
高流速的地方壓力小
低流速的地方壓力大
所以
A跟C點的壓力最大
B點壓力最小

English: 
Streamlines are drew around the sphere
and the green part represents a boundary layer
which is a important region
Let's study the region outside boundary layer first
where the Bernoulli's principle applies
Observe the motion of a small volume element
Since the speed of a baseball is much lower than sound speed
the compression effect can be ignore
and the volume of this small element keeps unchanged
Imagine it moves in a tube that continuously changes in diameter
When it moves toward A
its speed decreases
A is called the "stagnation point"
around which the fluid separates to both sides
The speed increases from A to B
and reach the highest value at B
After passing through B the speed decreases
and reach the minimum speed when it arrive at C
Bernoulli's principle states that
high speed corresponds to low pressure
and low speed corresponds to high pressure
Therefore, the pressure maximums occurs at A and C
and B has the minimum pressure

Chinese: 
有趣的地方是發生在下游的區域
從B到C壓力是逐漸變大的
邊界層在這樣的外部壓力變化下
如果這個壓力差足夠大的話
會發生邊界層往上游逆流的現象
一旦發生逆流
上游的邊界層就會被推擠離開球體表面
這種現象叫做「邊界層分離」
「分離點」大約會發生在B點附近
在分離點之後的流體非常的不穩定
會不停打轉
然後生出各種大大小小的渦流
這種情況稱之為「亂流」
因為它是出現在障礙物的後方
所以這整個亂流區域還有個特別的名子
叫做「尾流」
因為分離點就發生在B跟D點附近
所以尾流的寬度大致就和球體的寬度差不多
整體來說
尾流是一個低壓區域
這是造成空氣阻力的主要原因
然而分離點發生的位置
很容易因為障礙物的外形而變化
像是流線形的物體

English: 
Interesting thing occurs in the downstream region
The pressure gradually increases from B to C
Outside the boundary layer, when the external pressure variation is large enough
it can force the downstream boundary layer to move in opposite direction
Once the inverse flow occurs
upstream boundary layer can be pushed away from the surface
This phenomenon is called "Boundary layer separation"
The location of "Separation point" is near B
and the flow is very unstable behind the separation point
Round and round
When various scales of vortices emerge, it is called "Turbulence"
The tail-like turbulent region behind a obstacle is also named a "Wake"
Since the separation points occur close to B and D
the wake width is comparable to the diameter of the sphere
As a whole, wake is a low pressure region
and this is the main reason for air resistance
However, the location of separation points
are sensitive to the geometric shape of obstacles
For a streamlined body

Chinese: 
分離點會發生在尾端附近
尾流變小了
阻力也就大為縮減
這就是為什麼機翼
以及太陽能車要設計成流線型的原因
除了障礙物的外形
另一個決定分離點的因素是障礙物表面的粗糙程度
先來看一組有趣的實驗
左邊的圖是光滑球體的實驗
分離點大約發生在球體最大截面積的地方
右圖是在上游的地方加上一圈細線
結果分離點向後退
尾流變小了
這是怎麼回事呢？
為什麼球面上的微小凸起會產生這樣的變化？
原因就在於邊界層的狀態改變了
原本應該是穩定的「層流邊界層」
因為球面上的凸起部分製造出小尺度的渦流
使得邊界層演變成亂流的狀態
這稱之為「亂流邊界層」
我們知道邊界層外面的流體不受黏滯力的影響
所以有較大的流速

English: 
the separation occur near its tail
The smaller the wake width
the smaller the drag
This is the reason why airfoils and solar cars
are designed to be streamlined form
Besides the geometric shape of obstacles
the surface roughness also determines the location of separation points
Take a look at this interesting experiment
The left photo represents a smooth sphere in wind tunnel
and the separation points are close to the largest cross section of the sphere
The right photo shows that with a thin trip wire in the upstream
the separation points move backward and the wake shrinks
What's going on here?
Why a small raised surface leads to this result?
The reason is that the state of boundary layer has been changed
It is originally a laminar flow in the upstream boundary layer
however, since the raised surface creats small scale vortices
the boundary layer becomes turbulent
This is called a "Turbulent boundary layer"
Fluid flows faster outside the boundary layer because no viscous effect there

English: 
A turbulent boundary layer is stirred by vortices
which makes it mixed with the outer and faster fluid
Therefore, the average velocities of boundary layer increases
and get more momentum toward downstream
So that the separation points retreat and the wake shrinks
The purpose of dimple design on a golf ball
is to create a turbulent boundary layer and reduce the drag
Baseballs are not smooth spheres, too
The seam lines on the surface of a balseball can disturb boundary layer
In general, baseball seams can cause the retreat of separation points
Baseball seams are not uniformly distributed on the surface
and this could result in the wake deflection
Moreover, the rotations of a baseball are classified into two types
One is four-seamer: There are 4 seams passing by in one full rotation
The other one is two-seamer: There are 2 seams passing by in one full rotation

Chinese: 
亂流邊界層因為受到渦流的攪動
使得它與周圍較快的流體發生混合
這樣一來
邊界層的平均速度變快了
有更大的動量往下游衝擊過去
如此便造成分離點向後退
尾流也跟著縮小了
高爾夫球在表面上設計了很多的小凹洞
目的也是為了製造出亂流邊界層
以達到減少阻力的目的
然而
棒球也不是光滑球體
表面有約一公釐高的縫線會干擾邊界層
通常的情況
棒球的縫線也會導致分離點的位置延後
縫線在棒球表面的分布是不均勻的
這會使得尾流發生偏折的現象
然而
棒球的旋轉又可以分為兩種方式
一種是旋轉一圈的時候
有四條縫線劃過
稱之為「四縫線球」
另一種是旋轉一圈
有兩條縫線劃過
稱之為「二縫線球」

Chinese: 
攻角的變化會使棒球受到不對稱的作用力
這可以藉由風洞實驗加以測量
這張圖是代表四縫線球
在轉速等於零時
各種角度所產生的作用力
可以看到棒球自轉一周
作用力重覆變化四個週期
我們現在運用電腦動畫
更詳細的來說明這個實驗結果
空氣固定的由左邊流進來
一開始角度等於零的時候
縫線在兩側的分布是對稱的
所以尾流沒有偏折
位於球的正後方
一旦改變球的攻角
尾流很快的就發生偏折
在大約22度附近向上偏折最大
要如何解釋偏折的原因呢？
上游的邊界層氣流在經過縫線之後
會變成亂流邊界層

English: 
The changes in attack angle results in non-symmetric force
which can be measured by wind tunnel experiments
This figure shows the force versus attack angle
for a four-seamer at zero rotation speed
As you can see, the force has four periods of variation in one full rotation
The detail explanations for that figure are given in the following computer graphics
The air flows in steadily from the left
In the beginning the attack angle is zero
and the seams are symmetrically distributed
therefore, the wake is right behind the ball without being deflected
But the wake deflects dramatically when the attack angle changes
The largest upward deflection occurs when the attack angle is about 22 degree
What is the reason for wake deflection ?
The air in upstream boundary layer becomes turbulent after passing by the seams

Chinese: 
請注意下半邊的縫線比上半邊更接近停滯點
也就是下半邊的亂流邊界層提早發生
走過更長的路徑
有更多的機會發生混合
因此邊界層得到更大的速度與動量向下游流去
下半邊的分離點就延後了
相反的
上半邊的分離點較早
流速也較慢
兩側的邊界層氣流在尾流中混合後
平均速度偏向上方
這就是尾流發生偏折的原因
要理解棒球所受到的作用力
我們可以把中間複雜的交互作用過程當成黑盒子看待
原本水平流進來的空氣
經過黑盒子後向上偏折
這表示空氣受到向上的作用力
從牛頓第三運動定律可以知道
在這個黑盒子中
一定有大小相等
方向相反的反作用力
作用在棒球上
這就是縫線的不對稱分布所造成的結果

English: 
Notice the seam in the lower half part is more close to the stagnation point
and the lower half turbulent boundary leads the upper one
With longer path and more chance for mixing
the boundary layer gets more velocity and momentum moving downward
so the lower half part separation point retreats more
On the contrary
the upper half part separation point is in advance and the average velocity is lower
After mixing of these two sides of boundary layer flow in the wake
the average velocity deviates upward
and this is the explanation for wake deflection
To realize how the force acting on the ball
one can regard the complex interaction processes as a black box
The horizontal air flow deflects upward after passing through the black box
implies that a upward force acting on the air flow
Simultaneously, there must be a reaction force in the black box exert on the ball
which is equal in magnitude and opposite in direction
as specified by Newton's third law
This is the result of non-symmetric seam distribution

Chinese: 
繼續改變攻角
到達45度時作用力回復為零
再來尾流會向另一側偏折
到68度時作用力最大
90度時又回復為零
這樣子就走完一個週期
所以轉一圈360度作用力就有四個週期的變化
若是改成以二縫線的方式旋轉
實驗結果發現自轉一周有兩個大的週期變化
作用力的最大值相當於棒球重量的2/3
週期比四縫線大一倍
我們先把座標系定義好以方便之後的討論
使用直角座標系
X方向指向本壘板
Y方向指向一壘側
並且令XY平面與地面平行
Z軸垂直於地表
這樣定義的座標系
升力剛好在正Z方向
重力則在負Z方向
側向力在正負Y方向

English: 
Keep on varying the attack angle
the force vanishes at about 45 degree
beyond which the wake deflects to another side
and the force reachs its maximum at about 68 degree
Then the force vanishs again at 90 degree
that it has goes through a period
As a result, there are 4 periods in a full rotation of 360 degree
In the case of two-seam rotation for a knuckleball
there are two main periods of force variation
The maximum force is equivalent to 2/3 of baseball weight
and the period is two times larger than the case of four-seam rotation
Define the coordinate system first before follow-up discussions
Use the rectangular coordinate system
and let X axis pointing toward home plate
Y axis pointing toward first base
Let XY plane be parallel to the ground
so Z axis stands vertically
For this coordinate system
The lift force is in the positive Z direction
the gravity is in the negative Z direction
the lateral force is in the positive/negative Y direction

Chinese: 
阻力在負X方向
我們先假設蝴蝶球的自轉軸垂直於地表
從俯視圖觀察
假設飛行過程中球自轉了半圈
根據前面風洞實驗的結果來推斷
四縫線的蝴蝶球會有兩個週期的左右搖擺
改成二縫線旋轉的話
蝴蝶球的軌跡會有一個週期的變化
在實際應用上
投手必須要盡可能的壓抑球的轉動
所以投球時要利用指尖把球向前推出
這樣的投法是相當難以控制的
自轉軸不一定會固定在特定方向
再加上作用力的變化對於攻角相當敏感
這使得蝴蝶球的軌跡非常難以捉摸
在一次的練習當中
紅襪隊的威克菲爾為觀眾展示了蝴蝶球的漂移能力
從蝴蝶球的研究我們已經了解到

English: 
and the drag force is in the negative X direction
Now consider the rotation axis of a knuckleball is perpendicular to the ground
Observe the trajectory from top view
and assume the ball spins half a rotation during the flight
The wind tunnel experiment implies that
there are two periods of left and right motion for a four-seam knuckleball
In the case of two-seam rotation
the knuckleball trajectory has one period of variation
In practice
a pitcher would try to throw the ball as no spin as possible
The way is to push the ball with fingertip
but it makes the ball hard to be controlled
Since the spin axis may not be fixed in some specific direction
and since the force is sensitive to attack angle
the knuckleball trajectory becomes hard to be predicted
This is a demonstration of knuckleball's movement
by Tim Wakefield in a Japan TV show

Chinese: 
在極低轉速下
縫線的位置
或者說攻角
完全決定棒球的受力情形
但是在棒球運動中
其他種類的球路都比蝴蝶球自轉快10倍以上
在高速自轉的情況下還要考慮新的物理效應
那就是「馬格納斯力」
為了要排除縫線的干擾
我們以光滑的球體來說明
當球體開始自轉的時候
B側的表面跟周圍空氣的運動是順向的
C側剛好是反向
我們知道邊界層內的流體受到黏滯力控制
所以B側的邊界層會比C側具有較大的流速往下游流動
也就是B側的邊界層具有較大的動量
這樣
B側的分離點自然較為延後
兩側的氣體混合後造成尾流向下方偏移
球體因此受到橫向的作用力
這種作用力是由轉動所引起的

English: 
From the research of knuckleball we learn that at extremely low spin rate
the seam locations or say attack angle
determines the force on the baseball
In baseball games, however
other kinds of pitches have spin rate over 10 times higher than knuckleball
and a new physical effect of force enters in-
- the "Magnus force"
To get rid of the disturbance by seams
let's begin with a smooth sphere
When it start to spin
The B-side surface and the ambient air move in the same direction
while C-side moves reversely
Since the fluid inside boundary layer is governed by viscosity
the B-side boundary layer flows faster than that of C-side toward downstream
and carries more momentum
Therefore, the B-side separation point becomes relative backward
so the wake deflects downward after mixing of these two sides of air
and the transverse force arises
This is called the "Magnus effect"

English: 
that resulted from the rotation of moving object in the fluid
The magnitude of Magnus force is probably proportional to
the air flow speed V and the angular frequency ω of the ball
So the empirical formula for Magnus force is given by
1/2 times "Magnus coefficient CM"
"air density ρ"
"baseball cross-sectional area A"
"radius R"
"angular frequency ω" and the "flow speed V"
The Magnus coefficient is a dimensionless quantity
and is the only one parameter to be determined from experiments
Rotation is the motion characterize with direction
and can be well described by a "Vector"
For example, a spin vector S
Point your right-hand thumb in the direction of the arrow
and the grip of the other four fingers represents gyration of the object
In addition
the length of the arrow is used to represent the spin rate
Therefore, a vector can give a complete description of rotation

Chinese: 
叫做「馬格納斯效應」
馬格納斯力的大小
大致上正比於空氣的流速V
還有球體的自轉角頻率ω
所以馬格納斯力的經驗公式為
1/2乘以「馬格納斯係數CM」
「空氣密度ρ」
「棒球截面積A」
「半徑R」
「自轉角頻率ω」以及「空氣流速V」
馬格納斯係數是一個無因次的量
也是唯一的待定係數
可以從實驗測量得知它的值
任何物體的轉動是有方向性的
所以適合使用「向量」來加以描述
譬如說自旋向量S
把拇指指向箭頭方向
那麼四指握起來就代表物體迴旋的方向
更進一步的
還可以用箭頭的長短來代表轉速的快慢
所以向量這種東西可以完整的描述一個物體的自轉

Chinese: 
至於說馬格納斯力的方向
可以利用右手定則來決定
以四隻指頭指向棒球前進的方向
拇指指向自旋向量
這樣掌心面對的方向就代表馬格納斯力作用的方向
我們已經了解光滑球體轉動的馬格納斯效應
那如果把縫線的干擾加進來呢？
尾流的方向開始變得搖擺不定
球體的自轉
與表面的縫線產生綜合的效應
使得馬格納斯力變成隨時間變化
雖然說
邊界層複雜的行為還沒有被完全的研究透澈
然而
透過實驗的測量
還是可以得到平均的作用力
如果球的自旋向量水平
指向三壘的一側
這樣馬格納斯力就會貢獻在升力上面
這張圖是實驗測量結果

English: 
Now we can utilize the right-hand rule
to determine the direction of Magnus force
Point your four fingers toward the direction of ball flight
and align the thumb with spin vector S
then you got your palm faces the direction of Magnus force
We've learned the Magnus effect by a spinning smooth ball
What about a spinning ball with seams?
The wake becomes fluttering
The spin of the ball as well as the surface seams result in joint effect
and the Magnus force varies periodically
Although the complex behavior of boundary layer has not been well studied
the average force can still be obtained from experiments
If the spin vector points toward the third base horizontally
the Magnus force will totally contribute to lift
This figure is the measurement results

English: 
The horizontal axis represents the spin parameter SP
which is defined as the surface speed Rω of a rotating sphere
divided by the flow speed V
The vertical axis represents the lift coefficient CL
which is defined by Magnus coefficient times SP
The upper curve represents the lift coefficient of 4-seamer
and the lower one represents the lift coefficient of 2-seamer
Taking a 140 km/hr and 20 rev/sec fastball as an example
The lift coefficient for a 4-seamer is about two times larger than that of a 2-seamer
The difference between them decreases
with increasing the spin rate or decreasing the ball speed
In this 140kmh and 20rps case:
The lift for a 4-seamer is about 60% of the weight
while it is about 30% for a 2-seamer
Now the trajectory can be estimated since the lift force is known
Green and red curves denote the trajectories of 4-seamer and 2-seamer, respectively
And the dashed line is a straight line

Chinese: 
橫軸是自轉參數SP
定義為球體表面圓周運動速度Rω
除以流速V
縱軸是升力係數CL
定義為馬格納斯係數乘以SP
上面的曲線代表四縫線球的升力係數
下面的是二縫線球的升力係數
以一顆時速140公里
轉速每秒20轉的快速球為例
四縫線球的升力係數大約是二縫線球的兩倍
增加轉速
或者降低球速
兩者的差距會明顯的縮小
跟棒球的重量相比
這個四縫線球的升力相當於棒球重量的60%左右
而二縫線球則約為30%
知道升力的大小就可以估算運動軌跡
綠線與紅線分別代表四縫線與二縫線快速球的運動軌跡
虛線是一條直線

Chinese: 
當球底達本壘板
四縫線球大約會掉落40公分左右
而二縫線球大約掉落70公分左右
兩者差距約30公分
若是完全不考慮升力
在只受重力作用的情況下
棒球會下落接近1公尺的程度
另一種極端是
自轉軸垂直地面的情況
這樣馬格納斯力完全作用在側向力的方向上
以同樣的球速與轉速作為例子
當二縫線球在抵達本壘板時
會有大約30公分向左
或者向右的橫向移動
四縫線球則大約有60公分的橫向移動
在棒球的術語裡面
這種橫向或垂直移動的能力稱之為「尾勁」
由於表面粗糙程度的關係
棒球的阻力係數介於光滑球體與高爾夫球之間

English: 
When the ball arrives at the plate
the 4-seamer drops about 40 cm
while the 2-seamer drops about 70 cm
and the difference is about 30 cm
If there is no lift force
and consider only the gravitational force on the ball
the ball would drop about 1 meter
Another extreme case is a ball with vertical spin axis
such that Magnus force acts totally in the direction of lateral force
Consider the same spin rate and ball speed discussed above
A 2-seamer moves about 30 cm to the left or right
when it arrive at the plate
and the horizontal movement for a 4-seamer is about 60 cm
The ability of horizontal or vertical motion is called "tail-strength"
in Taiwan's baseball terminology
Owing to the surface roughness
the drag coefficient of a baseball is in between that of a smooth ball and a golf ball

Chinese: 
雖然說一般認為
二縫線球比四縫線球阻力大
但是從一般的實驗資料來看
兩者的差別並不大
而且
在特殊條件下
二縫線球的阻力反而還明顯的小於四縫線球
投四縫線快速球的時候
把食指跟中指按在縫線上
拇指放在正下方
投出去的時候手指往下扣
讓球產生這樣子的旋轉
四縫線快速球的自轉軸通常會有些傾斜
這會產生往右打者內側移動的尾勁

English: 
General speaking, people think that
2-seamers have larger drag than 4-seamers
But the experiment data show that difference is not big
On the contrary
2-seamers have obviously smaller drag for some special conditions
To pitch a 4-seam fastball
place your index and middle fingertips on the baseball seam
and place your thumb right beneath the ball
At release point, press the fingers downward
and get the ball backspin like this
The spin axis of a 4-seam fast ball is in general oblique
which results in the inside movement for a right-handed batter

Chinese: 
從投手的視角來看
自旋向量S指向右下
利用右手定則得知
馬格納斯力M指向右上
至於說重力則是指向下
做一個平行四邊形
就得到合成力
投卡特球的時候
食指與中指稍微偏向外側
放球時
手指往下扣
讓球產生這樣子旋轉

English: 
4-seam fastball
In the pitcher's view angle
Spin vector S pointing toward lower right
and the right-hand rule tells that
Magnus force M pointing toward the upper right
As for gravity, Fg pointing downward
The resultant force is obtained by making a parallelogram
To pitch a cutter
place the index and middle fingers a little bit outside
and press the fingers downward at release point
and get the ball spin like this
Cutter (Pitcher's view angle)

Chinese: 
投二縫線快速球的時候
食指與中指放在兩條縫線最靠近的地方
拇指放在正下方
投出後會以二縫線方式旋轉
如果食指稍微用力的話
讓球產生這樣子的旋轉
產生更大幅度的側移跟下沉
這就是所謂的伸卡球

English: 
To pitch a 2-seam fastball
place the index and middle fingertips on the narrow part of the seams
and place the thumb right beneath the ball
then it will rotate as a 2-seamer after delivery
2-seam fastball (Pitcher's view angle)
If push off the index finger at release point and let the ball side spin
the ball will get more lateral movement and sinking
and it becomes a "sinker"
2-seam sinker (Pitcher's view angle)

English: 
To pitch a slider
place the index and middle fingers outside of the ball
and rotate your palm a little bit
At release point, press the fingers downward
and let the ball spin in this way
Slider (Pitcher's view angle)
To pitch a curveball
rotate your palm to the left
At release point, rotate the fingers forward
and let the ball spin in this way

Chinese: 
投滑球的時候
食指與中指放在球的外側
手掌稍微的轉向側面
放球時手指往下扣
讓球產生這樣子的旋轉
投曲球的時候
手掌完全轉向側面
放球時手指向前轉動
讓球產生這樣子的旋轉

Chinese: 
投指叉球的時候
把食指與中指岔開
夾住球的兩側
拇指放在球的下方
這樣子投出去的球會有較低的轉速
產生大幅度的下沉
投變速球的時候
先把手比成OK的姿勢
然後把球放進來
這種握法會降低球的轉速

English: 
Another view angle for curveball
To pitch a forkball
Split your index and middle fingers apart to grip the ball
and place the thumb beneath the ball
It leads to low spin rate and large sink
Forkball (Pitcher's view angle)
To pitch a changeup
make an "OK" gesture
then put the ball in your hand
This results in low spin rate

Chinese: 
產生類似於指叉球的效果
縱向滑球是滑球的一種變形
它的自轉軸是平行於運動的方向
這樣的話馬格納斯力就變成零
只剩下重力跟阻力作用在棒球上
所以它會快速的向下墜落
子彈球的自轉軸介於縱向滑球與卡特球之間

English: 
and is similar to a forkball
Changeup (Pitcher's view angle)
Vertical slider is a special variant of slider
The Magnus force vanishes
since the spin axis aligns with its motion
Gravity and drag are the rest of forces acting on the ball
therefore, it drop fast vertically
The spin axis of a gyroball is in between that of V-slider and cutter

English: 
and so does its characteristics
It can move as fast as a cut fastball
and may also sinks like a V-slider
Nevertheless
since every pitch has wide range of physical characteristics
and since there are no standard criteria for classification
some people think that gyroball can just be classified into cutter or slider
To pitch a screwball
Turn your palm inside out to pitch the ball
The spin direction of a screwball for a right-handed pitcher
is similar to that of a left hander's slider or curve
so the pitch moves down and in on a right-handed batter
The screwball pitchers are rare
because it tends to damage pitcher's arms
Conclusions
We have gone into the detail of the physics of boundary layer

Chinese: 
同時也具有這兩種球路的特性
它有接近於快速球的球速
而且還如同縱向滑球一般
可以大幅度的下沉
不過
各種球路的特性範圍很寬
球種之間的差異性也沒有一定的標準可以當作區隔
所以也有人認為把子彈球歸類在卡特球或者滑球的類型即可
投球螺旋的時候
手掌往外翻
這樣子把球投出去
以右投來說
螺旋球的自轉方向與左投的滑球跟曲球相類似
所以它會向右打者的內側偏移與滑落
由於螺旋球非常的傷害手臂
所以這種投手非常稀少
我們花了很大的篇幅描述並且解釋邊界層的行為

Chinese: 
也介紹了各式各樣的變化球路
希望這些豐富的內容有助於增添平時各位看球
還有玩球的樂趣

English: 
and have leaned various baseball pitches
Hope you have fun watching and playing baseball games
