Game Development Reference
In-Depth Information
v
FF
v
=−
x
(5.17a)
x
D
v
FF
v
y
=−
(5.17b)
y
D
v
F
=−
g
−
F
v
z
(5.17c)
z
D
1
---
rv
2
AC
D
F
D
=
(5.17d)
Since drag force acts in the opposite direction to velocity, the general effect of drag is that
it reduces the projectile velocity. Looking at the z-direction force equation shown in Equation
(5.17c), when the projectile is climbing, drag will slow down the rate of climb. When the projectile
is falling back down to Earth, drag will slow the rate of fall.
The acceleration equations for the trajectory model with gravity and drag are obtained by
dividing the force equations in Equation (5.17) by the projectile mass,
m
. The acceleration
equations are shown in Equation (5.18).
Fv
a
=−
Dx
(5.18a)
x
mv
Fv
Dy
a
=−
(5.18b)
y
mv
Fv
a
=− −
g
Dz
(5.18c)
z
mv
Another significant difference between the gravity-only model and the equations of motion
when aerodynamic drag is included is that the accelerations are now a function of the projectile
mass. Specifically, the acceleration due to aerodynamic drag is inversely proportional to the mass
of the projectile. All other things being equal, a cannonball with a mass of 5
kg
will experience
twice the acceleration due to drag than will a cannonball with a mass of 10
kg
.
Velocity Equations
In determining the velocity components for a projectile when aerodynamic drag is included in
the model, we start once again with the differential equations that relate a velocity derivative
with respect to time to an acceleration. Using the acceleration equations shown in Equation (5.18),
the differential velocity equations for the projectile are as follows:
dv
F v
x
==−
a
D
x
(5.19a)
x
dt
mv
dv
F v
y
D
y
==−
a
(5.19b)
y
dt
mv
dv
F v
z
==−−
a
g
D
z
(5.19c)
z
dt
mv
