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Projectile Motion
Plot Scaling
100
25
Delay = 0.1
45
30
0.05
8000
0
1.191.54E-05
9.8
0.1
80.0 90.0 100.0
Prepared by G.W. O'Leary and R.J. Ribando
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Computed Data
Computed Variables Computed Resu
Rhobar 0.00014875 Time Position
Amass 1.00007438 X Y
Bgrav 9.79854225 (sec) (m) (m)
Ccoef 0.00223125 0.00 0.0000 0.0000
0.10 2.1178 2.0689
0.20 4.2287 4.03320.30 6.3329 5.8935
0.40 8.4307 7.6502
0.50 10.5221 9.3037
0.60 12.6074 10.8546
0.70 14.6867 12.3032
0.80 16.7602 13.6500
0.90 18.8281 14.8952
1.00 20.8905 16.0393
1.10 22.9475 17.0825
1.20 24.9993 18.0253
1.30 27.0460 18.8678
1.40 29.0876 19.61041.50 31.1244 20.2534
1.60 33.1564 20.7970
1.70 35.1836 21.2414
1.80 37.2062 21.5870
1.90 39.2242 21.8339
2.00 41.2376 21.9824
2.10 43.2465 22.0327
2.20 45.2509 21.9850
2.30 47.2509 21.8396
2.40 49.2463 21.5966
2.50 51.2373 21.2563
2.60 53.2238 20.8188
2.70 55.2058 20.2846
2.80 57.1832 19.6536
2.90 59.1560 18.9263
3.00 61.1241 18.1029
3.10 63.0875 17.1835
3.20 65.0462 16.1685
3.30 67.0000 15.0582
3.40 68.9488 13.8529
3.50 70.8926 12.5528
3.60 72.8313 11.1583
3.70 74.7648 9.6697
3.80 76.6930 8.0873
3.90 78.6158 6.41154.00 80.5330 4.6427
4.10 82.4446 2.7813
4.20 84.3504 0.8277
4.30 86.2504 -1.2178
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Computed Data
lts
Velocity
Horizontal Vertical
(m/s) (m/s)
21.2132 21.2132
21.1433 20.1651
21.0754 19.122121.0094 18.0840
20.9454 17.0506
20.8832 16.0217
20.8227 14.9969
20.7639 13.9762
20.7068 12.9592
20.6511 11.9459
20.5968 10.9360
20.5439 9.9294
20.4922 8.9259
20.4416 7.9253
20.3922 6.927520.3436 5.9324
20.2959 4.9398
20.2490 3.9498
20.2027 2.9621
20.1569 1.9767
20.1116 0.9936
20.0665 0.0126
20.0217 -0.9661
19.9770 -1.9426
19.9322 -2.9169
19.8874 -3.8890
19.8423 -4.8589
19.7969 -5.8264
19.7511 -6.7916
19.7048 -7.7543
19.6580 -8.7145
19.6105 -9.6720
19.5623 -10.6268
19.5133 -11.5788
19.4635 -12.5277
19.4128 -13.4736
19.3611 -14.4162
19.3085 -15.3555
19.2548 -16.2912
19.2001 -17.223319.1443 -18.1516
19.0874 -19.0760
19.0294 -19.9963
18.9702 -20.9124
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Sample Data for Alternative Projectiles
Type Mass Diameter Volume Density
(kg) (m) (m^3) (kg/m^3)
Beach Ball 0.0960 0.3800 0.0287309 3.341
Nerf Ball 0.0125 0.1050 0.0006061 20.623
Kickball 0.5630 0.2700 0.0103060 54.628
Ping Pong Ball 0.0023 0.0400 0.0000335 68.636
Soccer Ball 0.4370 0.2200 0.0055753 78.382
Basketball 0.5950 0.2400 0.0072382 82.202
Tennis Ball 0.0560 0.0650 0.0001438 389.448
Softball 0.1840 0.0950 0.0004489 409.872
Baseball 0.1440 0.0700 0.0001796 801.807
Water Balloon 0.5230 0.1000 0.0005236 998.856
Golf Ball 0.0460 0.0440 0.0000446 1031.338
Shotput 6.8100 0.1176 0.0008514 7999.030
All diameters and masses are approximate.
Most of these are not exactly smooth spheres, and some are deformable.
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Disclaimer
This collection of worksheets was developed for the
Session on Projectile Motion and Computer Modeling,
presented at the 1997 Summer Institute of the
Southeastern Consortium for Minorities in Engineering, Inc.
held at the University of Virginia June 15 - June 26, 1997.
It is based on Program 1.4 in An Introduction to Computational
Fluid Dynamics by Chuen-Yen Chow, Wiley (1979)
R.J.Ribando, 310 MEC, Univ. of Virginia, June 1997
Copyright 1997, All rights reserved.
This program may be distributed freely for instructional purposes
only providing that:
(1) The file be distributed in its entirety including disclaimer
and copyright notices.
(2) No part of it may be incorporated into any commercial product.
DISCLAIMER
The author shall not be responsible for losses of any kind
resulting from the use of the program or of any documentation
and can in no way provide compensation for any losses sustained
including but not limited to any obligation, liability, right,
or remedy for tort nor any business expense, machine downtime
or damages caused to the user by any deficiency, defect or
error in the program or in any such documentation or any
malfunction of the program or for any incidental or consequential
losses, damages or costs, however caused.
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Tech Details (1)
Some Technical Details (1)
If we are willing to ignore the effect of drag on the projectile, t
of a simple, spherical projectile simplify greatly - to the point thaqt we
solve them. But a computer or even a graphing calculator does provide
the solution.
For those cases involving uniform acceleration (which it will
air drag is neglected), the distance traveled is simply the average veloci
s ance = e oc y x meaverage
The average velocity is given by:
average initial
The acceleration is the change in velocity over the elapsed time (and is
cce era on = e oc y e ocfinal -
Solve this for the final velocity:
e oc y e oc y cce er final initial=
Combining the first, second and fourth equations:
Distance = Velocity x Timeinitial +
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Tech Details (1)
he equations that govern the flight
ont even need a computer to
a convenient means of visualizing
e shown later is appropriate when
ty times the elapsed time:
final
assumed uniform here):
y meinitial
a on x me
1
2Acceleration x Time2
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Tech Details (2)
Some Technical Details (2)
In order to determine the trajectory of our idealized spherical
Second Law:
F m a=
that is, the force is equal to the mass times the acceleration. Well incl
that is, the weight, but will ignore air drag for now. Forces and velociti
is, they have both magnitude and direction. (The state trooper is intere
magnitude of your velocity, but if you are trying to get somewhere in p
ell resolve forces (and accelerations and velocities) into components
(vertical) directions and apply Newtons 2nd
law separately to each.
Since we have ignored air drag, there are no forces in the x (h
horizontal acceleration is identically 0.0. That means the horizontal ve
equal to the initial value Uinitial . The horizontal position is then given
x meinitial initial= +
In the y (vertical) direction, we consider only the force due to
F ma mgy y= = - ,
that is, the acceleration in the vertical direction is equal to -g (9.8 m/s2
the English system. With this uniform acceleration, the vertical veloci
initial= - .
Finally the vertical position is given by:
Y Y V x Time1
2g Time
initial initial
2= + -
The initial velocity components specified in these equations can be fou
initia initia initia
initia initia initia
The equations for X and Y are easily input to a graphing calculator in t
trajectory can be visualized as a function of time, launch velocity (Velo
(Angleinitial).
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Tech Details (2)
projectile, well apply Newtons
de the force due to gravity here,
es are both vector quantities, that
sted in your speed, which is the
rticular, your velocity is key.)
in the x (horizontal) and y
rizontal direction), thus the
locity (U) will be constant and
by:
ravity:
in the metric system, 32.2 ft/s2
in
ty (V) is then given by:
d from simple trigonometry:
his parametric form so that the
cityinitial) and launch angle
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Tech Details (3)
Some Technical Details (3)
The model of projectile motion developed on the previous she
implementation on a graphing calculator, has some obvious problems.
consequence, we found that contrary to intuition, the horizontal velocit
never decreases. Furthermore, the vertical velocity just keeps getting
downward) with time; that is, it never reaches a terminal velocity. To
include the force due to the drag of the air on the spherical projectile.
will be more important for a light sphere, e.g., a beach ball, and less so
put.
The air drag model and the solution algorithm implemented i
explained inAn Introduction to Computational Fluid Dynamics by C.Y
few highlights are presented here. First of all, this is a 2-D model onl
allowed. The drag force depends on the velocity of the projectile relati
to have only a horizontal component and acts opposite to the relative wi
drag coefficient of asmooth sphere are used. This function Cdrag imp
The accelerations in the x and y directions at each point in time are co
and FYoverM, respectively. Unfortunately with the extra terms invol
governing equations cant be solved directly (they are a set of two non-l
equations). So we use a numerical technique calledRunge-Kutta integimplemented in the subroutine Kutta. All the heavy-duty calculation
FyoverM and the subroutine Kutta) were all implemented behind-the-s
pplications and are automatically invoked when the user hits the Com
In addition to the main sheet, which includes boxes for user in
graphically, another sheet reports the computed x and y positions and t
velocity components as a function of time. Another sheet gives some a
common spherical projectiles which the user may want to test.
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Tech Details (3)
et, while convenient for
Air drag was ignored and as a
y stays at its initial value and
ore and more negative (heading
rectify this problem we must
Our experience tells us that drag
for heavy projectiles like a shot
this spreadsheet are fully
. Chow, (Wiley, 1979). Only a
- no hooks, slices or curveballs
ve to the wind, which is assumed
nd. Experimental data for the
lements curve fits for this data.
puted in the functions FXoverM
ing the air drag, the two
inear, ordinary differential
ration which has been(the functions Cdrag, FxoverM,
enes in Visual Basic for
pute/Plot button on the main sheet.
put and shows the trajectory
e horizontal (u) and vertical (v)
proximate data for various