Physics 199BB The Physics of Baseball

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Physics 199BBThe Physics of Baseball

• Fall 2007 Freshman Discovery Course
• Alan M. Nathan
• 403 Loomis
• 333-0965
• a-nathan_at_uiuc.edu
• Week 3

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The Flight of a Baseball

• Recall from last time
• The goal to develop an understanding of the
  trajectory of a baseball in flight
• Pitched baseball
• Batted baseball
• Thrown baseball
• We solved for projectile motion with gravity
• by solving equations
• by using Excel
• We introduced the concept of drag

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Reminders

• To convert mph to ft/s, multiply by 1.467
• To convert ft/s to mph, divide by 1.467
• g 9.8 m/s2 32.2 ft/s2

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Recall also how to use Excel to compute the
trajectory

• divide up time into slices separate by dt
• dt needs to be small
• suppose x,y,vx,vy are known at time t
• at time tdt
• x(tdt)x(t)vx(t)dt
• y(tdt)y(t)vy(t)dt
• vx(tdt)vx(t)ax(t)dt
• vy(tdt)vy(t)ay(t)dt
• use values at t0 as starting point
• x0,y0,v0x,v0y
• ax0 ay-g

look at trajectory-v1.xls
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Lets compare with exact calculation

• Compare range D with exact
• D v02sin(2?)/g
• angle for maximum range
• sin(2?) is maximum at ?45o

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Baseball Trajectories with Drag

• Fdrag ½ CD?Av2
• ? is density of air
• A is cross sectional area of ball
• v speed of ball
• CD is drag coefficient
• Direction of force is exactly opposite velocity

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Finally recall that we estimated the size of the
drag force

• Let CD 0.5, v100 mph
• FD ½CD?Av2
• FD 0.574 lb
• By comparison, weight of ball is 5.1 oz
• mg 0.319 lb
• We conclude that the drag is very important

this is where we left off last time
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A useful numerical formula
FD ½CD?Av2 plug in know values of ? and A
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Direction of the Drag Force
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Using Excel to Compute the Trajectory

• Acceleration is no longer constant
• Divide into tiny time increments
• Assume over very short times, v and a are
  constant
• ?x vx?t ?vx ax?t
• ?y vy?t ?vy ay?t
• ay includes both gravity and drag
• ay 1.67 x 10-4 Cdvvyg g
• ax includes only drag
• ax 1.67 x 10-4 Cdvvxg

look at trajectory-v2-class.xls
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Some Observations

• Drag reduces distance significantly
• Drag reduces flight time
• Drag reduces maximum height
• Drag causes trajectory to be asymmetric
• falling angle steeper than rising angle
• Drag reduces angle for maximum distance

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Lets add a scroll bar

• http//www.rose-hulman.edu/moloney/BoiseSSWorksho
  p/Sliders/SlidersOnSpreadsheets.htm

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Applications

• Suppose Joel Zumaya tosses a pitch at 95 mph as
  it leaves his hand.
• What is the speed as it crosses home plate?
• What is the average speed?

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Approximate solution

• FD ½CD?Av2 ma m(vo-vf)/t
• mvo(1-vf/v0)/t
• t ? D/v0
• Plug in... (1-vf/v0) ½CD?AD/m
• Now put in numbers.
• (1-vf/v0) 0.00538DCD
• for D55 ft and CD0.35, (1-vf/v0) 0.10
• vf 0.90v0
• Excel gets 0.907vo

compare with Exceltrajectory-v2-class.xls
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Applications

• Suppose Joel Zumaya tosses a pitch at 95 mph as
  it leaves his hand.
• What is the speed as it crosses home plate?
• vf/vo ? 0.9
• What is the average speed?
• vave? 0.95 vo
• What does radar gun report?

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Applications

• 2. Compare vertical drop and time of arrival of
  90 mph fastball with 85 mph changeup.

Exceltrajectory-v2-class.xls
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Applications

• Compare vertical drop and time of arrival of 90
  mph fastball with 85 mph changeup.
• 85 mph drops 3 inches more and arrives 0.020 sec
  later
• Batter will be way out in front

Exceltrajectory-v2-class.xls
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Now we allow CD to vary
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Now we allow CD to vary
look at cd-class.xls and trajectory-v3-class.xls
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Applications

• 3. Suppose a ball is dropped from 500 ft. How
  fast is it going when it hits the ground?

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Applications

• 3. Suppose a ball is dropped from 500 ft. How
  fast is it going when it hits the ground?
• About 95 mph
• FD W

FD
W
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Homework

• Set up Excel for calculating trajectories with
  drag and our model for CD
• For v0 100 mph, find angle for maximum distance
• For ?35o, plot maximum distance for v0 in the
  range 50-120 mph

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Forces on a Baseball in Flight

• Gravity
• Already discussed
• Drag (air resistance) Force
• Already discussed
• Magnus Force
• Now we do this

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First some definitions

• ? is angular velocity
• a measure of how fast the ball is spinning
• units are rad/s or rev/min (rpm)
• to convert from rad/s to rpm
• multiply by 60/(2?)
• to convert from rpm to rad/s
• divide by 60/(2?)
• ? has a direction

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The spin axis is the line connecting the south to
north pole (right-hand rule)
spin axis
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The Magnus Force

• FM ½CL?Av2
• CL is the lift coefficient
• Force acts in the direction
• another right-hand rule
• force is perpendicular to both v and ?
• force acts in the direction that the leading edge
  of the ball is turning

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Magnus force causes ball to break in direction
that front edge of the ball is turning
Example straight overhand fastball as viewed by
batterupward break opposing gravity
deflection
spin axis
rotation
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More examples, as seen by batter
deflection
spin axis
overhand FB upward break
spin axis
12-6 overhand CB downward break
deflection
spin axis
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More examples, as seen by batter
Sidearm curveball by RPH breaks away from RHB
3/4 fastball by RHP breaks up and in to RHB
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More examples, as seen by batterNote that spin
axis is tilted a bit forward.
cut fastball by RHP breaks up and away to RHB
Last slide
slider by RHP breaks down and away to RHB