Game Development Reference
This kind of equation is called a “differential equation.” It links the different dif-
ferentials together, sometimes with the original quantity: in this case the second dif-
p and the original p . Differential equations can sometimes be solved to give
an expression for just the original quantity. In our case the equation can be solved to
give us an expression that links the position with the current time. 2 The expression is
solved to give
p t =
p 0 cos (χ t)
sin (χ t)
where p 0 is the position of the end of the spring relative to the natural length at the
start of the prediction, and
p 0 is the velocity at the same time.
We can substitute into equation 6.4 the time interval we are interested in (i.e., the
duration of the current frame), and work out where the spring would end up if it
were left to do its own thing. We can then create a force that is just big enough to get
it to the correct location over the duration of the frame. If the final location needs to
be p t , then the force to get it there would be
and the acceleration
p is given by
p 0 ) 1
( p t −
t 2 − ˙
Note that, although this gets the particle to the correct place, it doesn't get it there
with the correct speed. We'll return to the problems caused by this failing at the end
of the section.
Damped Harmonic Motion
A real spring experiences drag as well as spring forces. The spring will not continue
to oscillate forever to the same point. Its maximum extension will get less each time,
until eventually it settles at the rest length. This gradual decrease is caused by the drag
that the spring experiences.
When we run our physics engine normally, the drag will be incorporated in the
damping parameter. When we predict the behavior of the spring using the previous
formula this does not happen.
2. Not all differential equations have a simple solution, although most simple equations of the preceding
kind do. Solving differential equations can involve applying a whole range of different techniques and is
beyond the scope of this topic. When necessary I will provide the answers needed for the physics simulator.
If you want to understand more about how I get these answers, you can refer to any undergraduate-level
calculus text for more details.