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 2

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The Flight of a Baseball

• The goal to develop an understanding of the
  trajectory of a baseball in flight
• Pitched baseball
• Batted baseball
• Thrown baseball
• First step we need to go over some basic
  physics concepts

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Position, Velocity, and Acceleration

• Position x,y,z
• Units of length (m, ft, )
• Trajectory completely known if we know the
  position of an object at every instant of time
  x(t),y(t),z(t)
• Position is a vector with three components

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Position, Velocity, and Acceleration

• 2. Velocity vx,vy,vz
• Units of length/time (m/s, ft/s,mph,)
• Velocity is the rate of change of position
• Velocity is a vector with three components

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Position, Velocity, and Acceleration

• 3. Acceleration ax,ay,az
• Units of length/time2 (m/s2, ft/s2,)
• Acceleration is the rate of change of velocity
• Acceleration is a vector with three components

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Motion with Constant Acceleration

• x x0 v0t ½at2
• v v0 at

Special Case 1 a0
x x0 v0t v v0
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Special Case 2Two-dimensional projectile motion
with gravity
ax 0 (horizontal)ay -g (vertical)g
32.2 ft/s2 9.8 m/s2

• x x0 v0xt vx vx0 (constant)
• y y0 v0yt - ½gt2 vy v0y - gt

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Detailed example pitched baseball

• Suppose a pitcher throws a baseball with an
  initial horizontal velocity of 90 mph at a height
  of 6 ft above home plate. How long does the
  pitch take to reach home plate? How much does
  the pitch drop vertically?

x x0 v0xt x00 v0x 90 mph y y0
v0yt - ½gt2 y06 ft v0y0
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Detailed example pitched baseball
x x0 v0xt x00 v0x 90 mph y y0
v0yt - ½gt2 y06 ft v0y0
Want to find time T when x 60.5 ft. Use that T
to find y. But firstneed a consistent set of
units. Convert mph to f/s 90 mile/hour 90
(mile/hour)(5280 ft/mile)(1/3600
hour/sec) 901.467132.0 ft/s
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Useful thing to remember

• To convert mph to ft/s, multiply by 1.467
• To convert ft/s to mph, divide by 1.467

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Detailed example pitched baseball
x x0 v0xt x00 v0x 90 mph y y0
v0yt - ½gt2 y06 ft v0y0
Now solve to find T 60.5 ft 0 132T
ft/s T0.458 s Now solve to find y(T) Y 6 ft
0 -0.532.2 ft/s2(0.458)2 s2 6 ft 3.382
ft 2.618 ft Ball drops 3.4 ft!
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Using Excel to Compute the Trajectory

• divide up time into slices separate by dt
• 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
• for case at hand values known at t0
• x0,y0,v0x,v0y
• ax0 ay-g

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Detailed example batted baseball

• Suppose the baseball is hit at an initial height
  of 3 ft off the ground at a speed of 100 mph and
  an angle of 35o to the horizontal.
• How far does it travel?
• How long is it in the air?
• How high does it go?

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Detailed example batted baseball

• Suppose the baseball is hit at an initial height
  of 3 ft off the ground at a speed of 100 mph and
  an angle of 35o to the horizontal.

v0 100 mph 146.7 ft/s ? 35o v0x v0
cos(?) v0y v0 sin(?)
y
v0
?
x
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Batted Ball Example

• x x0 v0xt x0 v0tcos(?)
• y y0 v0tsin(?) - ½gt2
• x00 y03 ft v0146.7 ft/s ?35o

First step How long T is ball in the
air? Trick when ball hits ground,
vy-v0y-v0sin(?) Use vyv0y-gT and solve for T,
with vy-voy. T2v0y/g2v0sin(?)/g2146.7sin(350
)/32.25.23 s
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Batted Ball Example

• x x0 v0xt x0 v0tcos(?)
• y y0 v0tsin(?) - ½gt2
• x00 y03 ft v0146.7 ft/s ?35o

Second step How far D did ball travel? Use D
v0Tcos(?) 146.75.23cos(35o) 628.5 ft
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Batted Ball Example

• x x0 v0xt x0 v0tcos(?)
• y y0 v0tsin(?) - ½gt2
• x00 y03 ft v0146.7 ft/s ?35o

Third step How high H did ball go? Maximum
height occurs at time tT/2 2.62 s. Plug into
equation for y, using tT/2 H 6
146.72.62sin(35o)-0.532.2(2.62)2 115.9 ft
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Now lets use Excel to solve(just like before)

• 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
• for case at hand values known at t0
• x0,y0,v0x,v0y
• ax0 ay-g

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Some Useful Formulas(we wont use these for
anything)

• Maximum distance D v02sin(2?)/g
• Maximum height H v02sin2(?)/2g
• Time of flight T 2v0sin(?)/g
• D is largest when ?45o
• T and H are largest with ?90o

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Baseball Trajectories with Drag and Magnus Forces

• Some additional physics concepts
• Newtons First Law
• Objects at rest stay at rest and objects in
  motion continue to move at constant velocity if
  not acted upon by an external force
• In other words, with no external force v is
  constant in both magnitude and direction
• Newtons Second Law
• Forces cause acceleration a F/m or

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Forces on a Baseball in Flight

• Gravity
• Already discussed
• Drag (air resistance) Force
• We will do this next
• Magnus Force
• We will do this later

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Baseball Trajectories with Drag

• Fdrag ½ CD?Av2
• ? is density of air
• 1.23 kg/m3 at normal temp and pressure
• A is cross sectional area of ball
• A ?R2 4.16 x 10-3 m2
• v speed of ball
• CD is drag coefficient
• A number between 0 and 1
• Approximately 0.5 for vlt50 mph
• See plot in Adair, p. 8, Fig. 2.1
• Direction of force is exactly opposite velocity

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Drag Coefficient from Adair
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Lets estimate size of drag force

• Let CD ½, v100 mph
• FD ½CD?Av2
• Convert mph to m/s 100 mph 44.7 m/s
• FD1/21/21.234.16x10-3(44.7)2
• FD 2.56 N 0.574 lb
• By comparison, weight of ball is 5.1 oz
• mg 0.319 lb
• We conclude that the drag is very important

last slide of Week 2