Projectile Motion (Two Dimensional)

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Projectile Motion(Two Dimensional)

• Accounting for Drag

2
Learning Objectives

• Know the equation to compute the drag force on an
  object due to air friction
• Apply Newton's Second Law and the relationship
  between acceleration, velocity and position to
  solve a two-dimensional projectile problem,
  including the affects of drag.
• Prepare an Excel spreadsheet to implement
  solution to two-dimensional projectile with drag.

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Projectile Problem - No Drag
V0
y
Position
q
x
Velocity Acceleration Vx Vocos(q) ax
0 Vy Vosin(q) - g t ay -g
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Projectile Problem - Drag

• All projectiles are subject to the effects of
  drag.
• Drag caused by air is significant.
• Drag is a function of the properties of the air
  (viscosity, density), projectile shape and
  projectile velocity.

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General Drag Force

• The drag FORCE acting on the projectile causes it
  to decelerate according to Newton's Law
• aD FD/m
• where FD drag force
• m mass of projectile

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Drag Force Due to Air

• The drag force due to wind (air) acting on an
  object can be found by
• FD 0.00256 CDV2A
• where FD drag force (lbf)
• CD drag coefficient (no units)
• V velocity of object (mph)
• A projected area (ft2)

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Pairs Exercise 1

• As a pair, take 3 minutes to convert the
  proportionality factor in the drag force equation
  on the previous slide if the
• units of velocity are ft/s, and
• the units of area are in2

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Drag Coefficient CD

• The drag coefficient is a function of the shape
  of the object (see Table 10.4).
• For a spherical shape the drag coefficient ranges
  from 0.1 to 300, depending upon Reynolds Number
  (see next slide).
• For the projectile velocities studied in this
  course, drag coefficients from 0.6 to 1.2 are
  reasonable.

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Drag Coefficient for Spheres
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Projectile Problem - Drag

• Consider the projectile, weighing W, and
  travelling at velocity V, at an angle q.

• The drag force acts opposite
• to the velocity vector, V.

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Projectile Problem - Drag

• The three forces acting on the projectile are
• the weight of the projectile
• the drag force in the x-direction
• the drag force in the y-direction

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Drag Forces

• The total drag force can be computed by
• FD 8.264 x 10-6 (CD V2 A)
• where
• V2 Vx2 Vy2

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Drag Forces

• The X and Y components of the drag force can be
  computed by
• FDx -FD cos(q)
• FDy -FD sin(q)
• where q arctan(Vy/Vx)

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Pair Exercise 2

• Derive equations for ax and ay from FDx and FDy.
• Assuming ax and ay are constant during a brief
  instant of time, derive equations for Vx and Vy
  at time ti knowing Vx and Vy at time ti-1 .
• Assuming Vx and Vy are constant during a brief
  instant of time, derive equations for x and y at
  time ti knowing x and y at time ti-1 .

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PAIRS EXERCISE 3

• Develop an Excel spreadsheet that describes the
  motion of a softball projectile
• 1) neglecting drag and
• 2) including drag

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PAIRS EXERCISE 3 (cont)

• Plot the trajectory of the softball (Y vs. X)
• assuming no drag
• assuming drag
• Answer the following for each case
• max. height of ball
• horizontal distance at impact with the ground

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Data for Pairs Exercise 3

• Assume the projectile is a softball with the
  following parameters
• W 0.400 lbf
• m 0.400 lbm
• Diameter 3.80 in
• Initial Velocity 100 ft/s at 30o
• CD 0.6
• g 32.174 ft/s2 (yes, assume you are on planet
  Earth)

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Hints for Pairs Exercise 3

• Reminder for the AES
• F ma/gc
• where gc 32.174 (lbm ft)/(lbf s2)
• The equations of acceleration for this problem
  are
• ax (FDx )gc/m
• ay (FDy -W)gc/m

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Considerations for Pairs Exercise 3

• What is a reasonable Dt ?
• What happens to the direction of the drag force
  after the projectile reaches maximum height?

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Sample Excel Spreadsheet
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Sample Chart