Transcript Mathematics of Baseball

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=mxv
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
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 1
mi
hr
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.
mi
hr
The change in the ball’s momentum?
The average force on the bat?
Ball’s momentum
P = momentum
Momentum: mass x velocity
mi
p( t ) - p( t ) = (5oz x 110 hr ) - (5oz x 90
mi
oz x mi
p= 5oz x 20 hr = 100 hr
1
0
– Usually unit for momentum is
will be oz x mi
kg x m
s
mi
hr
)
but in our case, it
hr
The change in momentum over the time
that the bat touches the ball is 100 ozhrx mi
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 kg x m
s²
1
t1  t0
.001
x
 f (t )
0
According to Newton’s Second Law
F=mxa
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)
.001

5oz x 20
0
1hr
.001  0
mi
hr 2
oz x mi
hr ²
.001
mi
1hr
0 5oz x 20 hr 2 =.001  0
= .1
Average force 100
oz x mi
x .1 hr ²
oz x mi
oz x mi
= Change of momentum 100
hr
hr
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

Kinetic Energy cont…
Equation: KE = 1 m x v²
2
m = mass
v = velocity
Kinetic Energy is usually measured in
Joules
2
1
kg
x
mi
1 Joule =
hr 2
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
1kg x mi2
Usual unit for Kinetic Energy is hr 2
In our case, we will use oz x mi 2
hr 2
Example 2
What is the work required for the
pitcher to throw a 90 fastball of a
5oz ball?
mi
hr
Work required to move the ball from s0
s1?
s = position
Work in Moving the Ball
What is the work done in moving the ball from s0
to s1?
Work = Kinetic energy
W =  f ( s ) ds
s1
s
s1 0

f ( s ) ds = ½ m x(v ) 2- ½ m x (v ) 2
0
1
s0
– 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
s1

s0
2
f ( s ) ds= 20250 oz mi
hr 2
20250 Joules
Bibliography
http://www.colloquial.com/games/baseball/math.ht
ml
http://en.allexperts.com/q/Calculus2063/2009/1/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.sciencehouse.org/student/bw/sports/projectile.html