Game Development Reference
In-Depth Information
Tidbit
Cannons have been used in warfare since the early 14th century. Early cannons fired hand-cut
stone balls because it was too expensive to smelt iron cannonballs. Cannons were used to fire large arrows
as well as cannonballs. As they became more sophisticated and reliable, cannons revolutionized warfare on
land and sea, making it possible to easily destroy castle walls and other fortifications.
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In your game programming simulations, you can make the size and mass of the cannon-
ball whatever you like. Early stone cannonballs sometimes weighed as much as 400 pounds.
The English and French typically used cannonballs weighing 4, 8, 12, 16, and 24 pounds. The
muzzle velocity of a cannonball was a function of the size of the cannonball and how much
gunpowder was used to fire the cannon. The English culverin (a midsized cannon) would fire
its ball with a muzzle velocity of about 260
m/s
(865
ft/s
). An upper end to the muzzle velocity
for a cannonball is the speed of sound. At a temperature of 21
o
C
(70
o
F
) the speed of sound is
344
m/s
(1129
ft/s
).
Arrows
When an archer pulls back on a bowstring, he is performing work that is converted to potential
energy and stored in the elastically deformed bow (also called a bowstave). When the bowstring
is released, the potential energy stored in the bow is converted into kinetic energy as the bow
snaps back to its undeformed position. If an arrow is in contact with the bowstring when it is
released, some of the potential energy is transferred to the arrow, causing it to fly away.
A bow is essentially a type of spring. The potential energy,
E
P
, stored in the bow is a function
of the force,
F
, required to draw the bowstring back and the distance,
x
, that the bowstring
was pulled.
1
2
P
Ee
=
x
(5.33)
The
e
term in Equation (5.33) is an efficiency coefficient that accounts for the fact that the
bow does not behave exactly like an idealized spring, meaning that not all of the work done on
the bow by bending it will be stored as potential energy. Wood bows typically have an efficiency
factor of about 0.9, whereas modern composite bows can have an efficiency factor greater
than one.
When the bow is released, the arrow flies off and acquires a certain kinetic energy. As the
bow is also moving back to its original position, it acquires a certain kinetic energy as well. The
potential energy in the deformed bow therefore is transferred to the kinetic energy of both the
arrow and the bow.
1
1
1
2
2
eFx
=
m v
+
k
m v
(5.34)
aa
ba
2
2
2
In Equation (5.34),
m
a
is the mass of the arrow,
v
a
is the velocity of the arrow,
m
b
is the
mass of the bow, and
k
is a scaling factor that accounts for the fact that different parts of the bow
move at different velocities when the bow is released. The value of
k
depends on the geometry
and material properties of the bow. Typical values of
k
for wood bows range from 0.03 to 0.07.
