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