Fig. 5 shows a perfect elastic collision of two bodies.
In reality, upon collision small elasticity effects
store the kinetic energy over a short period of time and return
the stored energy during a contiguous short interval of time. Because
the time scale of this phenomenon is very small compared to the
behavior of interest, it can be modeled as an instantaneous change
at a point in time governed by Newton's collision rule
and conservation of momentum ( 
)
which yields
(4) 
This algebraic relation only holds at a point in time.
Therefore, the switching specifications have
to ensure that the collide mode is departed immediately after
the state vector is updated, 
. Mathematically this is
represented as
(5) 
where 
is a pinnacle.
This shows that switching specifications have to be in terms of
a priori state variable values as shown by Fig. 5.
Figure 5: A collision between two bodies.