A Calculus Project By:
This presentation is the property of its rightful owner.
Sponsored Links
1 / 14

A Calculus Project By: PowerPoint PPT Presentation


  • 88 Views
  • Uploaded on
  • Presentation posted in: General

A Calculus Project By: Matt Jaffe, Eli Greenwald, Harry Brownstein, Sarah Eisenstark and Jake Starr.

Download Presentation

A Calculus Project By:

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


A calculus project by

A Calculus Project By:

Matt Jaffe, Eli Greenwald, Harry Brownstein, Sarah Eisenstark and Jake Starr


A calculus project by

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).


Question 1

Question 1

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


Prove that p t 1 p t 0 f t dt

Question 1, Section (a)

Prove that P(t1)- P(t0)= F(t) dt

P(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 calculus project by

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)


Find the change in the ball s momentum

Question 1, Section (b), Subsection (i)

Find the change in the ball’s momentum.

ρ = (m2v2) –(m1v1)

ρ = (49.1744 × .015)-(40.2336× .015)

ρ = .134112 kg m/s


Find the average force on the bat

Question 1, Section (b), Subsection (ii)

Find the average force on the bat:

J=FnetT=ΔP

ΔP= .134112

T=.001s

Fnet= 134.112N


A calculus project by

Question 2

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.


S how that f s ds mv 1 2 mv 2 2

Question 2, Section (a)

Show that ∫F(s) ds =½mv12- ½mv22

F(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)

-


How many foot pounds of work does it take to throw a baseball at a speed of 90 mi h

Question 2, Section (b)

How many foot-pounds 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 foot-pounds


A calculus project by

Question 3, 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.)


Question 3 section a

Question 3, Section (a)

=  =  =

To anti-derivative

Ln v = + c  v = e-t/10 + c  100 = e0 + ec

V= e-t/10 (100) v= -1000e-t/10+ cd=-1000e-t/10 + c

0 = -1000e0 + cc = 1000  280 = -1000e-t/10 + 1000

-1000 -1000


Question 3 section a continued

Question 3, Section (a) continued

-720=-1000e-t/10

-1000 -1000

.72 = e-t/10

ln.72 = -t/10

-.33 = -t/10

t = 3.3s


A calculus project by

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….30-17


  • Login