Game Development Reference
In-Depth Information
Figure 6-13. Magnus effect sample program
Variable Mass
In Chapter 1 we mentioned that some problems in dynamics involve variable mass.
We'll look at variable mass here since it applies to self-propelled projectiles such as
rockets. When a rocket is producing thrust to accelerate, it loses mass (fuel) at some
rate. When all of the fuel is consumed (burnout), the rocket no longer produces thrust
and has reached its maximum speed. After burnout you can treat the trajectory of the
rocket just as you would a non-self-propelled projectile, as discussed earlier. However,
while the rocket is producing thrust, you need to consider its mass change since this
will affect its motion.
In cases where the mass change of the object under consideration is such that the mass
being expelled or taken in has 0 absolute velocity—like a ship consuming fuel, for ex‐
ample—you can set up the equations of motion as you normally would, where the sum
of the forces equals the rate of change in momentum. However, in this case mass will
be a function of time, and your equations of motion will look like this:
ΣF = m a = d/dt (m v) =m (dv/dt) + (dm/dt) v
You can proceed to solve them just as you would normally, but keep in mind the time
dependence of mass.
A rocket, on the other hand, expels mass at some nonzero velocity, and you can't use
the preceding approach to properly account for its mass change. In this case, you need
to consider the relative velocity between the expelled mass and the rocket itself. The
linear equation of motion now looks like this:
ΣF = m dv/dt + dm/dt u