A Calculus Project By: Matt Jaffe, Eli Greenwald, Harry Brownstein, Sarah Eisenstark and Jake Starr.
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Matt Jaffe, Eli Greenwald, Harry Brownstein, Sarah Eisenstark and Jake Starr
In this project we explore three of the many applications of calculus in baseball. The physical interactions of the game, especially the collision of the ball and bat, are quite complex and their models are discussed in detail in a book by Robert Adair, The Physics of Baseball, 3d ed. (New York: HarperPerennial, 2002).
It may surprise you to learn that the collision of baseball and bat lasts only about a thousandth of a second. Here we calculate the average force on the bat during this collision by first computing the change in the ball’s momentum.
The momentum p of an object is the product of its mass m and its velocity v, that is, p = mv. Suppose an object, moving along a straight line, is acted on by a force F = F(t) that is a continuous function of time
Question 1, Section (a) calculus in baseball. The physical interactions of the game, especially the collision of the ball and bat, are quite complex and their models are discussed in detail in a book by Robert Adair,
Prove that P(t1) P(t0)= F(t) dtP(t1) – P(t0) = ∫ F(t) dt
F(t) = ma(f)
mv(t1) – mv(t0) = ∫ ma(t) dt = m ∫ a(t) dt
= ( ) ( )
m∫ a(t) dt = m(v(t1))m(v(t0))
= P(t1)  P(t0)
A pitcher throws a 90 mi/h fastball to the batter, who hits a line drive directly back to the pitcher. The ball is in contact with the bat for 0.001 seconds and leaves the bat with velocity of 110 mi/h. The baseball weighs 5 oz, and in U.S. customary units, its mass is measured in slugs: m= w/g where g= 32 ft/s2
Question 1, Section (b)
Question 1, Section (b), Subsection ( a line drive directly back to the pitcher. The ball is in contact with the bat for 0.001 seconds and leaves the bat with velocity of 110 mi/h. The baseball weighs 5 oz, and in U.S. customary units, its mass is measured in slugs: m= w/g where g= 32 ft/si)
Find the change in the ball’s momentum.ρ = (m2v2) –(m1v1)
ρ = (49.1744 × .015)(40.2336× .015)
ρ = .134112 kg m/s
Question 1, Section (b), Subsection (ii) a line drive directly back to the pitcher. The ball is in contact with the bat for 0.001 seconds and leaves the bat with velocity of 110 mi/h. The baseball weighs 5 oz, and in U.S. customary units, its mass is measured in slugs: m= w/g where g= 32 ft/s
Find the average force on the bat:J=FnetT=ΔP
ΔP= .134112
T=.001s
Fnet= 134.112N
Question 2 a line drive directly back to the pitcher. The ball is in contact with the bat for 0.001 seconds and leaves the bat with velocity of 110 mi/h. The baseball weighs 5 oz, and in U.S. customary units, its mass is measured in slugs: m= w/g where g= 32 ft/s
In this problem we calculate the work required for a pitcher to throw a 90 mi/h fastball by first considering kinetic energy. The Kinetic energy of an object of mass m and velocity v is given by K= ½ mv2. Suppose an object of mass m moving in a straight line is acted on by forceF = F(s) that depends on its position s. according to Newton's Second Law:
F(s)= ma m
Where a and v denote the acceleration and velocity of the object.
Question 2, Section (a) a line drive directly back to the pitcher. The ball is in contact with the bat for 0.001 seconds and leaves the bat with velocity of 110 mi/h. The baseball weighs 5 oz, and in U.S. customary units, its mass is measured in slugs: m= w/g where g= 32 ft/s
Show that ∫F(s) ds =½mv12 ½mv22F(s) = ma= m(dv/dt) =mv(dv/ds)
v0= v(s0)
v1 = v(s1)
m∫ v dv
F(s) = ½mv2 from S(0) to (s1)

Question 2, Section (b) a line drive directly back to the pitcher. The ball is in contact with the bat for 0.001 seconds and leaves the bat with velocity of 110 mi/h. The baseball weighs 5 oz, and in U.S. customary units, its mass is measured in slugs: m= w/g where g= 32 ft/s
How many footpounds of work does it take to throw a baseball at a speed of 90 mi/h?F= ½ Mvr2 – ½ Mv02
M = .1417
V0 = 90 = 40.233
V1 = 0
F = ½ (.1417)(0)2 – ½ (.1417)(40.233)2
F = 114.68449
F = 114.69 footpounds
Question 3 a line drive directly back to the pitcher. The ball is in contact with the bat for 0.001 seconds and leaves the bat with velocity of 110 mi/h. The baseball weighs 5 oz, and in U.S. customary units, its mass is measured in slugs: m= w/g where g= 32 ft/s, Section (a)
An outfielder fields a baseball 280 ft away from home plate and throws it directly to the catcher with an initial velocity of 100 ft/s. Assume that the velocity v(t) of the ball after t seconds satisfies the differential equation dv/dt = 1/10 because of air resistance. How long does it take for the ball to reach home plate? (Ignore any vertical motion of the ball.)
= = =
To antiderivative
Ln v = + c v = et/10 + c 100 = e0 + ec
V= et/10 (100) v= 1000et/10+ cd=1000et/10 + c
0 = 1000e0 + cc = 1000 280 = 1000et/10 + 1000
1000 1000
720=1000et/10
1000 1000
.72 = et/10
ln.72 = t/10
.33 = t/10
t = 3.3s
That was the calculus of Baseball. I hope that you all think about this when you watch baseball from now on.The Indians are the best team in baseball as of now….3017