Projectile Motion

Prepared by G.W. O'Leary and R.J. Ribando

Computer Modeling of Projectile Motion

Frequent User InputsLaunch Angle (degrees) 45Launch Velocity (m/s) 30Diameter of Projectile (m) 0.05Density of Projectile (kg/m^3) 8000Fluid (Wind) Velocity (m/s) 0

Less Frequent User InputsDensity of Fluid (Air) (kg/m^3) 1.19Kinematic Viscosity of Fluid (Air) (m^2/s) 1.54E-05Acceleration of Gravity (m/s^2) 9.8Timestep (sec) 0.1

0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0

-5.0

0.0

5.0

10.0

15.0

20.0

25.0

Location of Spherical Projectile

X-Position (m)

Y-P

os

itio

n (

m)

Projectile Motion

Prepared by G.W. O'Leary and R.J. Ribando

Plot Scaling

100

25

Delay = 0.1

Computed Data

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Computed Variables Computed ResultsRhobar 0.000149 Time Position VelocityAmass 1.000074 X Y Horizontal VerticalBgrav 9.798542 (sec) (m) (m) (m/s) (m/s)Ccoef 0.002231 0.00 0.0000 0.0000 21.2132 21.2132

0.10 2.1178 2.0689 21.1433 20.16510.20 4.2287 4.0332 21.0754 19.12210.30 6.3329 5.8935 21.0094 18.08400.40 8.4307 7.6502 20.9454 17.05060.50 10.5221 9.3037 20.8832 16.02170.60 12.6074 10.8546 20.8227 14.99690.70 14.6867 12.3032 20.7639 13.97620.80 16.7602 13.6500 20.7068 12.95920.90 18.8281 14.8952 20.6511 11.94591.00 20.8905 16.0393 20.5968 10.93601.10 22.9475 17.0825 20.5439 9.92941.20 24.9993 18.0253 20.4922 8.92591.30 27.0460 18.8678 20.4416 7.92531.40 29.0876 19.6104 20.3922 6.92751.50 31.1244 20.2534 20.3436 5.93241.60 33.1564 20.7970 20.2959 4.93981.70 35.1836 21.2414 20.2490 3.94981.80 37.2062 21.5870 20.2027 2.96211.90 39.2242 21.8339 20.1569 1.97672.00 41.2376 21.9824 20.1116 0.99362.10 43.2465 22.0327 20.0665 0.01262.20 45.2509 21.9850 20.0217 -0.96612.30 47.2509 21.8396 19.9770 -1.94262.40 49.2463 21.5966 19.9322 -2.91692.50 51.2373 21.2563 19.8874 -3.88902.60 53.2238 20.8188 19.8423 -4.85892.70 55.2058 20.2846 19.7969 -5.82642.80 57.1832 19.6536 19.7511 -6.79162.90 59.1560 18.9263 19.7048 -7.75433.00 61.1241 18.1029 19.6580 -8.71453.10 63.0875 17.1835 19.6105 -9.67203.20 65.0462 16.1685 19.5623 -10.62683.30 67.0000 15.0582 19.5133 -11.57883.40 68.9488 13.8529 19.4635 -12.52773.50 70.8926 12.5528 19.4128 -13.47363.60 72.8313 11.1583 19.3611 -14.41623.70 74.7648 9.6697 19.3085 -15.35553.80 76.6930 8.0873 19.2548 -16.29123.90 78.6158 6.4115 19.2001 -17.22334.00 80.5330 4.6427 19.1443 -18.15164.10 82.4446 2.7813 19.0874 -19.07604.20 84.3504 0.8277 19.0294 -19.99634.30 86.2504 -1.2178 18.9702 -20.9124

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 3.341Nerf Ball 0.0125 0.1050 0.0006061 20.623Kickball 0.5630 0.2700 0.0103060 54.628Ping Pong Ball 0.0023 0.0400 0.0000335 68.636Soccer Ball 0.4370 0.2200 0.0055753 78.382Basketball 0.5950 0.2400 0.0072382 82.202Tennis Ball 0.0560 0.0650 0.0001438 389.448Softball 0.1840 0.0950 0.0004489 409.872Baseball 0.1440 0.0700 0.0001796 801.807Water Balloon 0.5230 0.1000 0.0005236 998.856Golf Ball 0.0460 0.0440 0.0000446 1031.338Shotput 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.

Disclaimer

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This collection of worksheets was developed for theSession 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 ComputationalFluid Dynamics by Chuen-Yen Chow, Wiley (1979)

R.J.Ribando, 310 MEC, Univ. of Virginia, June 1997Copyright 1997, All rights reserved.

This program may be distributed freely for instructional purposesonly 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.

DISCLAIMERThe author shall not be responsible for losses of any kindresulting from the use of the program or of any documentationand can in no way provide compensation for any losses sustainedincluding but not limited to any obligation, liability, right,or remedy for tort nor any business expense, machine downtimeor damages caused to the user by any deficiency, defect orerror in the program or in any such documentation or anymalfunction of the program or for any incidental or consequentiallosses, damages or costs, however caused.

Tech Details (1)

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Some Technical Details (1)

If we are willing to ignore the effect of drag on the projectile, the equations that govern the flightof a simple, spherical projectile simplify greatly - to the point thaqt we don’t even need a computer tosolve them. But a computer or even a graphing calculator does provide a convenient means of visualizingthe solution.

For those cases involving uniform acceleration (which it will be shown later is appropriate whenair drag is neglected), the distance traveled is simply the average velocity times the elapsed time:

Distance = Velocity x Timeaverage

The average velocity is given by:

Velocity = (Velocity Velocityaverage initial final ) / 2

The acceleration is the change in velocity over the elapsed time (and is assumed uniform here):

Acceleration = (Velocity Velocity Timefinal initial ) /

Solve this for the final velocity:

Velocity Velocity Acceleration x Timefinal initial

Combining the first, second and fourth equations:

Distance = Velocity x Time 1

2 Acceleration x Timeinitial

2

Tech Details (1)

Page 7

Some Technical Details (1)

If we are willing to ignore the effect of drag on the projectile, the equations that govern the flightof a simple, spherical projectile simplify greatly - to the point thaqt we don’t even need a computer tosolve them. But a computer or even a graphing calculator does provide a convenient means of visualizingthe solution.

For those cases involving uniform acceleration (which it will be shown later is appropriate whenair drag is neglected), the distance traveled is simply the average velocity times the elapsed time:

Distance = Velocity x Timeaverage

The average velocity is given by:

Velocity = (Velocity Velocityaverage initial final ) / 2

The acceleration is the change in velocity over the elapsed time (and is assumed uniform here):

Acceleration = (Velocity Velocity Timefinal initial ) /

Solve this for the final velocity:

Velocity Velocity Acceleration x Timefinal initial

Combining the first, second and fourth equations:

Distance = Velocity x Time 1

2 Acceleration x Timeinitial

2

Tech Details (2)

Page 8

Some Technical Details (2)

In order to determine the trajectory of our idealized spherical projectile, we’ll apply Newton’sSecond Law:

F m a

that is, the force is equal to the mass times the acceleration. We’ll include the force due to gravity here,that is, the weight, but will ignore air drag for now. Forces and velocities are both vector quantities, thatis, they have both magnitude and direction. (The state trooper is interested in your speed, which is themagnitude of your velocity, but if you are trying to get somewhere in particular, your velocity is key.)We’ll resolve forces (and accelerations and velocities) into components in the x (horizontal) and y(vertical) directions and apply Newton’s 2nd law separately to each.

Since we have ignored air drag, there are no forces in the x (horizontal direction), thus thehorizontal acceleration is identically 0.0. That means the horizontal velocity (U) will be constant andequal to the initial value Uinitial . The horizontal position is then given by:

X X U x Timeinitial initial

In the y (vertical) direction, we consider only the force due to gravity:

F ma mgy y ,

that is, the acceleration in the vertical direction is equal to -g (9.8 m/s2 in the metric system, 32.2 ft/s2 inthe English system. With this uniform acceleration, the vertical velocity (V) is then given by:

V V g x Timeinitial .

Finally the vertical position is given by:

Y Y V x Time 1

2 g Timeinitial initial

2

The initial velocity components specified in these equations can be found from simple trigonometry:

U Velocity x Cosine(Angle )initial initial initial

V Velocity x Sine(Angle )initial initial initial

The equations for X and Y are easily input to a graphing calculator in this parametric form so that thetrajectory can be visualized as a function of time, launch velocity (Velocityinitial) and launch angle(Angleinitial).

Tech Details (2)

Page 9

Some Technical Details (2)

In order to determine the trajectory of our idealized spherical projectile, we’ll apply Newton’sSecond Law:

F m a

that is, the force is equal to the mass times the acceleration. We’ll include the force due to gravity here,that is, the weight, but will ignore air drag for now. Forces and velocities are both vector quantities, thatis, they have both magnitude and direction. (The state trooper is interested in your speed, which is themagnitude of your velocity, but if you are trying to get somewhere in particular, your velocity is key.)We’ll resolve forces (and accelerations and velocities) into components in the x (horizontal) and y(vertical) directions and apply Newton’s 2nd law separately to each.

Since we have ignored air drag, there are no forces in the x (horizontal direction), thus thehorizontal acceleration is identically 0.0. That means the horizontal velocity (U) will be constant andequal to the initial value Uinitial . The horizontal position is then given by:

X X U x Timeinitial initial

In the y (vertical) direction, we consider only the force due to gravity:

F ma mgy y ,

that is, the acceleration in the vertical direction is equal to -g (9.8 m/s2 in the metric system, 32.2 ft/s2 inthe English system. With this uniform acceleration, the vertical velocity (V) is then given by:

V V g x Timeinitial .

Finally the vertical position is given by:

Y Y V x Time 1

2 g Timeinitial initial

2

The initial velocity components specified in these equations can be found from simple trigonometry:

U Velocity x Cosine(Angle )initial initial initial

V Velocity x Sine(Angle )initial initial initial

The equations for X and Y are easily input to a graphing calculator in this parametric form so that thetrajectory can be visualized as a function of time, launch velocity (Velocityinitial) and launch angle(Angleinitial).

Tech Details (3)

Page 10

Some Technical Details (3)

The model of projectile motion developed on the previous sheet, while convenient forimplementation on a graphing calculator, has some obvious problems. Air drag was ignored and as aconsequence, we found that contrary to intuition, the horizontal velocity stays at its initial value andnever decreases. Furthermore, the vertical velocity just keeps getting more and more negative (headingdownward) with time; that is, it never reaches a terminal velocity. To rectify this problem we mustinclude the force due to the drag of the air on the spherical projectile. Our experience tells us that dragwill be more important for a light sphere, e.g., a beach ball, and less so for heavy projectiles like a shotput.

The air drag model and the solution algorithm implemented in this spreadsheet are fullyexplained in An Introduction to Computational Fluid Dynamics by C.Y. Chow, (Wiley, 1979). Only afew highlights are presented here. First of all, this is a 2-D model only - no hooks, slices or curveballsallowed. The drag force depends on the velocity of the projectile relative to the wind, which is assumedto have only a horizontal component and acts opposite to the relative wind. Experimental data for thedrag coefficient of a smooth sphere are used. This function Cdrag implements curve fits for this data.The accelerations in the x and y directions at each point in time are computed in the functions FXoverMand FYoverM, respectively. Unfortunately with the extra terms involving the air drag, the twogoverning equations can’t be solved directly (they are a set of two non-linear, ordinary differentialequations). So we use a numerical technique called Runge-Kutta integration which has beenimplemented in the subroutine Kutta. All the heavy-duty calculations (the functions Cdrag, FxoverM,FyoverM and the subroutine Kutta) were all implemented behind-the-scenes in Visual Basic forApplications and are automatically invoked when the user hits the Compute/Plot button on the main sheet.

In addition to the main sheet, which includes boxes for user input and shows the trajectorygraph ically, another sheet reports the computed x and y positions and the horizontal (u) and vertical (v)velocity components as a function of time. Another sheet gives some approximate data for variouscommon spherical projectiles which the user may want to test.

Tech Details (3)

Page 11

Some Technical Details (3)

The model of projectile motion developed on the previous sheet, while convenient forimplementation on a graphing calculator, has some obvious problems. Air drag was ignored and as aconsequence, we found that contrary to intuition, the horizontal velocity stays at its initial value andnever decreases. Furthermore, the vertical velocity just keeps getting more and more negative (headingdownward) with time; that is, it never reaches a terminal velocity. To rectify this problem we mustinclude the force due to the drag of the air on the spherical projectile. Our experience tells us that dragwill be more important for a light sphere, e.g., a beach ball, and less so for heavy projectiles like a shotput.

The air drag model and the solution algorithm implemented in this spreadsheet are fullyexplained in An Introduction to Computational Fluid Dynamics by C.Y. Chow, (Wiley, 1979). Only afew highlights are presented here. First of all, this is a 2-D model only - no hooks, slices or curveballsallowed. The drag force depends on the velocity of the projectile relative to the wind, which is assumedto have only a horizontal component and acts opposite to the relative wind. Experimental data for thedrag coefficient of a smooth sphere are used. This function Cdrag implements curve fits for this data.The accelerations in the x and y directions at each point in time are computed in the functions FXoverMand FYoverM, respectively. Unfortunately with the extra terms involving the air drag, the twogoverning equations can’t be solved directly (they are a set of two non-linear, ordinary differentialequations). So we use a numerical technique called Runge-Kutta integration which has beenimplemented in the subroutine Kutta. All the heavy-duty calculations (the functions Cdrag, FxoverM,FyoverM and the subroutine Kutta) were all implemented behind-the-scenes in Visual Basic forApplications and are automatically invoked when the user hits the Compute/Plot button on the main sheet.

In addition to the main sheet, which includes boxes for user input and shows the trajectorygraph ically, another sheet reports the computed x and y positions and the horizontal (u) and vertical (v)velocity components as a function of time. Another sheet gives some approximate data for variouscommon spherical projectiles which the user may want to test.