Mathematics of Baseball

Jamie Kim

Kelsey Udinski

Clarissa Toy

Dixon 5*

What is baseball?

Two teams with nine players each on the field

Team A tries to deliver the ball to Team B where a Team B player tries to hit it

Teams score one run each time a batter moves around all four bases

Once a team accumulates three outs, the teams’ positions switch to either offense or defense

Baseball cont…

Nine turns on defense and nine or eight turns on offense

Typically nine innings: one turn on offense and one on defense

No time limit

Can’t result in a tie which may lead to extra innings

Momentum

A vector: size and direction

Any object in motion is said to have momentum

Size of momentum: the mass multiplied by the velocity

Measured in kg x m/s

The direction of the momentum: the direction of the object's velocity

Momentum cont…

P = m x vp: momentum of the objectm: mass of the objectv: velocity of the object

Mass: a fundamental measure of the amount of matter in the objectVelocity: how fast an object is moving to which direction

Average velocity can be calculated by dividing displacement over time V= ∆d ∆t

Application to Baseball

What is the average force exerted on the bat when the bat and the ball collide?

The time the bat actually touches the ball is only about .001 second

Example 1A pitcher throws a 90 ball to the batter who hits it directly back. The ball is in contact with the bat for .001 seconds and leaves the bat with the velocity of 110 . The ball weighs 5 oz.

The change in the ball’s momentum?

The average force on the bat?

hr

mi

hr

mi

Ball’s momentum

P = momentumMomentum: mass x velocity

p( ) - p( ) = (5oz x 110 ) - (5oz x 90 )

p= 5oz x 20 = 100– Usually unit for momentum is but in our case, it

will be

The change in momentum over the time that the bat touches the ball is 100

hr

mixoz

hr

mi

s

mxkg

hr

mixoz

hr

mixoz

hr

mi

hr

mi1t 0t

Average Force on the Bat

Average Force = total amount of force exerted during the time period divided by the time period

Usual unit for force is

x

According to Newton’s Second LawF = m x a

m = massa = acceleration = rate of change of velocity

– Derivative of velocity = v’(t)

F = mass x rate of change of velocity– Force units = oz x mi/hr/hr

Force = rate of change of momentum– Momentum = mass x velocity

Average force= 1/(t1-t0) x integral of F(t) dt over time (t0, t1)

= .1

= x .1

Average force 100 = Change of momentum 100

01

1

tt 001.

0

)(tf

²s

mxkg

001.

02

205hr

mixoz

²hr

mixoz

0001.

1

hr

001.

02

205hr

mixoz

²hr

mixoz0001.

1

hr

hr

mixoz

hr

mixoz

Kinetic Energy

the energy of motionthe work needed to accelerate a body of a given mass from rest to its current velocity

Three forms of kinetic energyvibrational rotational Translational

Kinetic Energy cont…

Equation: KE = 1 m x v² 2

m = massv = velocity

Kinetic Energy is usually measured in Joules

1 Joule = 2

21

hr

mixkg

Application to Baseball

What is the total work it takes for a pitcher to throw a ball?

The velocity of the ball is constantly changing during the time it leaves the pitcher’s hand and touches the bat

Usual unit for Kinetic Energy isIn our case, we will use

2

21

hr

mixkg

2

2

hr

mixoz

Example 2

What is the work required for the pitcher to throw a 90 fastball of a 5oz ball?

Work required to move the ball from

?s = position

hr

mi

0s

1s

Work in Moving the Ball

What is the work done in moving the ball from s0 to s1? Work = Kinetic energy

W =

= ½ m x - ½ m x– v(1) = v(s1) = velocity at position 1– v(0) = v(s0) =velocity at position 0

Force according to position = mass x acceleration5oz x

Kinetic Energy

= 2025020250 Joules

dssfs

s1

0

)(

dssfs

s1

0

)( 20 )(v2

1)(v

2

2

hr

miozdssfs

s1

0

)(

Bibliography

http://www.colloquial.com/games/baseball/math.html http://en.allexperts.com/q/Calculus-2063/2009/1/Calc-5.htm#b http://official-rules.org/overview/index.htm http://library.thinkquest.org/10796/ch2/ch2.htm http://www.stewartcalculus.com/data/ESSENTIAL%20CALCULUS%20Early%20Transcendentals/upfiles/projects/ess_wp_0705_stu.pdf http://www.science-house.org/student/bw/sports/projectile.html