Game Development Reference

In-Depth Information

velocity increases, so does drag since drag is a function of velocity. At some speed the

drag force retarding the object's motion will increase to a point where it is equal to the

gravitational force that's pulling the object toward the earth. In the absence of any other

forces that may affect motion, the net acceleration on the object is 0, and it continues

its descent at the constant terminal velocity.

Let us illustrate this further. Go back to the formula we derived for the
y
component

(vertical component) of velocity for the projectile modeled in the
Cannon2
example.

Here it is again so you don't have to flip back to
Chapter 4
:

v
y2
= (1 / C
d
) e
(-C
d
/m)t
(C
d
v
y1
+ m g) - (m g) / C
d

It isn't obvious from looking at this equation, but the velocity component,
v
y2
, asymptotes

to some constant value as time increases. To help you visualize this, we've plotted this

equation, as shown in
Figure 6-11
.

Figure 6-11. Terminal velocity

As you can see, over time the velocity reaches a maximum absolute value of about

−107.25 speed units. The negative velocities indicate that the velocity is in the negative

y-direction—that is, the object is falling toward the earth in this case. (For this calcu‐

lation we arbitrarily assumed a mass of 100, a drag coefficient of 30, and an initial velocity

of 0.)

Assuming an initial velocity of 0 and equating the formula for total resistance shown

earlier to the weight of an object, you can derive the following formula for terminal

velocity for an object in free fall:

2
mg

C
d
ρA

v
t
=