Game Development Reference
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v 2 = (T/C) - e -(C/m) t (T/C - v 1 )
where v 1 is the initial ship speed (which is constant) and v 2 is the ship speed at time t .
v 2 is what you're after here, since it tells you how fast the ship is traveling at any instant
of time.
Now that you have an equation for speed as a function of time, you can derive an
equation for displacement (distance traveled, in this case) as a function of time. Here,
you'll have to recall the formula v dt = ds , substitute the previous formula for speed,
integrate, rearrange, and solve for distance traveled. These steps are shown here:
v dt = ds
v 2 dt = ds
((T/C) - e -(C/m) t (T/C - v 1 )) dt = ds
(0 to t) (T/C) - e -(C/m) t (T/C - v 1 ) dt = ∫ (s1 to s2) ds
(T/C) ∫ (0 to t) dt - (T/C - v 1 ) ∫ (0 to t) e -(C/m) t dt = s 2 - s 1
[(T/C) t + ((T/C) - v 1 )(m/C) e -(C/m) t ] (0 to t) = s 2 - s 1
[(T/C) t + ((T/C) - v 1 )(m/C) e -(C/m) t ] - [0 + ((T/C) - v 1 )(m/C)] =
s 2 - s 1
(T/C) t + (T/C - v 1 ) (m/C) e -(C/m) t - (T/C - v 1 ) (m/C) = s 2 - s 1
s 2 = s 1 + (T/C) t + (T/C - v 1 ) (m/C) e -(C/m) t - (T/C - v 1 ) (m/C)
Finally you can write an equation for acceleration by going back to the original equation
of motion and solving for acceleration:
T - (C v) = m a
a = (T - (C v)) / m
where:
v = v 2 = (T/C) - e -(C/m) t (T/C - v 1 )
In summary, the equations for velocity, distance traveled, and acceleration are as follows:
v 2 = (T/C) - e -(C/m) t (T/C - v 1 )
s 2 = s 1 + (T/C) t + (T/C - v 1 ) (m/C) e -(C/m) t - (T/C - v 1 ) (m/C)
a = (T - (C v)) / m
To illustrate the motion of the ship further, we've plotted the ship's speed, distance trav‐
eled, and acceleration versus time, as shown in Figure 4-2 , Figure 4-3 , and Figure 4-4 .
To facilitate these illustrations, we've assumed the following:
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