Hi,
I recently wrote a text version of the old DOS Artillery Duel or Atari's Cannon Duel game. In this version, you play against the computer and can control the projectile's angle and velocity.

I'd now like to add one more dimension whereby the height of the cannons randomly changes for each game instantiation. (For example, the human's cannon might be at a 40 foot elevation while the computer's is at 10 feet).

The main calculation for the projectile is in the shot.awk script below. I've also included the other two routines in this game for anyone interested.

cannon.ksh - main program written in Korn Shell. Computer and human take turns shooting at one another

gettarget.awk - a simple awk script to get the target location (in this case the target is your computer enemy)

shot.awk - this is the routine (immediately below) where the distance is calculated based upon angle and velocity; and returns the cannon shot (to the main routine, cannon.ksh). I originally got the formulas and a basic understanding of the physics from forums like this one. However, I do not understand how to introduce the additional variable of elevation into the equation. I also don't understand many of the advanced notations and symbols used in formulas that do have a height variable. If you could use the word rather than the symbol (as you see coded in shot.awk) it would help me immensly.

From what I've read, I think I might have to introduce time into the equation. I can see how this would be required if I were plotting a point (on the screen) for graphics; however, I'm not sure it's necessary for this text version where we just print out how close the shot was to the target.

Thanks if you can help! -MB

shot.awk
---------
BEGIN {

# cannon.ksh passes in user-supplied angle and velocity
angle=ARGV[1]
ARGV[1]=""
velocity=ARGV[2]
ARGV[2]=""

gravity=(9.8*-1)

# convert angle to degrees
angle=(angle*3.14159265)/180

# original vector for x and y
vx=velocity*cos(angle)
vy=velocity*sin(angle)

# Midflight (velocity=0, no rise, no fall)
mf=(vy*-1)/gravity

# apex - used for graphics. Here for reference only
#apex=(vy*mf)+(0.5*gravity*(mf*mf))

# Fullflight is total time of flight
ff=mf*2

distance=vx*ff

printf("%3.0f\n",distance)
}

cannon.ksh
-----------
#! /bin/ksh

currentshot=0
keepshooting=1

# Computer variables
comp_angle=45
comp_hi=100.00
comp_low=1
currentguess=50.00

target=`awk -f gettarget.awk`

# Set level of difficulty
echo " "
echo "Select a level of difficulty: "
echo " "
echo "(e)asy: Displays both computer and human parameters"
echo "(m)edium: Displays only human parameters"
echo "(h)ard: Displays only + or - for human"
echo "(v)ery hard: Same as hard but does not even show you the distance to enemy"
echo " "
if [ "\$difficulty" != "v" ]
then
echo " "
echo "Target distance: \$target"
fi
echo " "
echo "Input angle (0-90) and velocity (1-100)"
echo " "

while [ \$keepshooting -eq 1 ]
do
currentshot=`expr \$currentshot + 1`

#
# Human shot
#

echo " "

angle=`echo \$parms | awk ' { print \$1 } '`
velocity=`echo \$parms | awk ' { print \$2 } '`

distance=`awk -f shot.awk \$angle \$velocity`

if [ \$distance -eq \$target ]
then
echo " "
echo " " # <--- this is Control-G to make the computer BEEP
echo " "
echo " "
echo " "
banner "BOOM!" # <--- banner is an old Unix utility to display the text in larger banner font
echo " "
echo "You hit the target at \$distance feet in \$currentshot shots! "
echo "Computer was using \$comp_angle \$currentguess"
echo " "
keepshooting=0
else

echo " "

if [ \$distance -gt target ]
then
delta=`expr \$distance - \$target`
if [ "\$difficulty" = "e" ] || [ "\$difficulty" = "m" ]
then
echo "+\${delta}"
else
if [ "\$difficulty" = "h" ] || [ "\$difficulty" = "v" ]
then
echo "+ you overshot"
fi
fi
else
delta=`expr \$target - \$distance`
if [ "\$difficulty" = "e" ] || [ "\$difficulty" = "m" ]
then
echo "-\${delta}"
else
if [ "\$difficulty" = "h" ] || [ "\$difficulty" = "v" ]
then
echo "- you undershot"
fi
fi
fi

fi

#
# Computer shot
#

# Only take a shot if the human didn't just hit us
if [ \$keepshooting -eq 1 ]
then
distance=`awk -f shot.awk \$comp_angle \$currentguess`

if [ \$distance -eq \$target ]
then
echo " "
echo " "
echo " "
echo " "
echo " "
banner "BAM!"
echo " "
echo "The computer got you at \$comp_angle \$currentguess in \$currentshot shots! Distance: \$target"
echo " "
keepshooting=0
else

echo " "

if [ \$distance -gt target ]
then
delta=`expr \$distance - \$target`
echo " "

if [ "\$difficulty" = "e" ]
then
echo "Computer: \$comp_angle \$currentguess +\${delta}"
else
echo "Computer missed!"
fi

# low routine from guess
comp_hi=\$currentguess
# need awk for fractions
currentguess=`echo \$currentguess \$comp_low | awk ' { printf("%2.2f",(\$1+\$2)/2 ) } ' `

else
delta=`expr \$target - \$distance`
echo " "
if [ "\$difficulty" = "e" ]
then
echo "Computer: \$comp_angle \$currentguess -\${delta}"
else
echo "Computer missed!"
fi

# hi routine from guess
comp_low=\$currentguess
# need awk for fractions
currentguess=`echo \$currentguess \$comp_hi | awk ' { printf("%2.2f",(\$1+\$2)/2 ) } ' `

# Check for 99.99 (in order to guess 100) - must convert to int
tcurrentguess=`echo \$currentguess | awk ' { printf("%d",\$1*100) } ' `
if [ \$tcurrentguess -eq 9999 ]
then
currentguess=100
fi
fi

fi
fi

done

exit

gettarget.awk
-------------
BEGIN {
trand=srand()
target=int(1000*rand())+1
printf("%d\n",target)
}

I've not checked your sources, but you need to work on a 2D coordinate system if you're taking elevation into account.
Your cannon fires from a set point, and the parabolic trajectory starts from there. If that point is translated vertically, the entire trajectory is translated vertically by the same amount (and extrapolated at the far end of course).

If you wanted to get real fancy you'd have to take things like changes in gravitational pull with increased altitude, changes in air pressure (and thus drag) at different altitudes, and curvature of the earth into account, but if you design your physics engine for flexibility those could be added later.

Cannon on a hill, lobbing the shot:
distance=vx*(vy/gravity+(sqrt((vy*vy)/(gravity*gravity)-2*height/gravity)))

Steep angle shot:
<snip> vx=velocity*cos(angle)
vy=velocity*sin(angle)

time=tgt_distance/vx

height=vy*time-.5*gravity*(time*time)

Steep angle downhill shot:

vx=velocity*cos(angle)
vy=velocity*sin(angle)

# For downhill tight shot, vertical velocity is negative
vy = vy * -1

time=tgt_distance/vx

height=vy*time-.5*gravity*(time*time)