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Mathematics of Baseball Jamie Kim Kelsey Udinski Clarissa Toy Dixon 5*

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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

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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

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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

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Momentum cont… P = m x v p: momentum of the object m: mass of the object v: velocity of the object Mass: a fundamental measure of the amount of matter in the object Velocity: how fast an object is moving to which direction Average velocity can be calculated by dividing displacement over time V= ∆d ∆t

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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

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Example 1 A 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?

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Ball’s momentum P = momentum Momentum: 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

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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 Law F = m x a m = mass a = 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

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Kinetic Energy the energy of motion the work needed to accelerate a body of a given mass from rest to its current velocity Three forms of kinetic energy vibrational rotational Translational

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Kinetic Energy cont… Equation: KE = 1 m x v² 2 m = mass v = velocity Kinetic Energy is usually measured in Joules 1 Joule =

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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 is In our case, we will use

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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

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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 acceleration 5oz x Kinetic Energy = 20250 20250 Joules

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Bibliography http://www.colloquial.com/games/baseball/math.ht ml 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/up files/projects/ess_wp_0705_stu.pdf http://www.science- house.org/student/bw/sports/projectile.html

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