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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)


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)


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

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