Transcript Design Process - Refinements

Projectile Motion
(Two Dimensional)
Accounting for Drag
Learning Objectives

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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.
Projectile Problem - No Drag
Position:
V0
y
x(t )  V0 cos( )t
1 2
y (t )  V0 sin(  )t  gt
2
Velocity:
Vx = Vocos()

x
Acceleration:
ax = 0
Vy = Vosin() - g t
ay = -g
Projectile Problem - Drag
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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.
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
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)
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
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
units of velocity are ft/s, and
the units of area are in2
Drag Coefficient: CD
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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.
Drag Coefficient for Spheres
Projectile Problem - Drag


Consider the
projectile,
weighing W,
and travelling at
velocity V, at an
angle .
FD
V
+y
+x
W
The drag force acts opposite
to the velocity vector, V.

Theta
Projectile Problem - Drag

The three forces acting on the projectile are:
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the weight of the projectile
the drag force in the x-direction
the drag force in the y-direction
FDy= FD sin(theta)

FDx= FDcos(theta)

+y
+x
Drag Forces

The total drag force can be computed by:
FD = 8.264 x 10-6 (CD V2 A)
where:
|V2|= Vx2 + Vy2
Drag Forces

The X and Y components of the drag
force can be computed by:
FDx = -FD cos()
FDy = -FD sin()
where:  = arctan(Vy/Vx)
Pair Exercise 2
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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 .
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 (con’t)
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Plot the trajectory of the softball (Y vs. X)
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assuming no drag
assuming drag
Answer the following for each case:
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max. height of ball
horizontal distance at impact with the ground
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Data for Pairs Exercise 3
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Assume the projectile is a softball with
the following parameters:
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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
More
planet Earth)
Hints for Pairs Exercise 3
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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
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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
Sample Chart