1.8 Continuous, Discrete, and Hybrid Models
13
Fig. 1.6 Graph of the position of the bouncing ball above the floor
After a rebound, the ball gains the speed v + = k · v − , where v + is the speed at
which the new flight begins, v − is the speed at which the ball touched the surface,
and k is a coefficient depending on the properties of the rebound. With an absolutely
elastic rebound, k = 1, with an absolutely inelastic k = 0, with an elastic 0 < k < 1.
We are interested in the moments of time at which the bounces occur and the law of
change in the height of the ball above the floor, as shown in Fig. 1.6.
When using the model of continuous and discrete time, this problem can be solved
only as a sequence of individual tasks. The processes occurring at the time of the
rebound can be neglected, that is, assume that the rebound occurs instantly. With this,
the rebound phenomenon itself is replaced by the instantaneous action v + = k · v − .
The first formulation of the problem: continuous time model.
In this simple problem, the solution of equations for each individual flight (except
the first) can be obtained in the explicit form:
h(t) = v 0 · t −
g·t
2
2
,
v(t) = v 0 − g · t,
where v 0 is the ball speed after rebound. Then, the next bounce time T can be found
from:
h(T ) = 0 ⇒ T =
2 · v 0
g
.
Consistently solving the equations of flight after the next bounce, we obtain the
required points in time.
The second formulation of the problem: discrete time model.
We use the solution obtained in order to solve a slightly more “complicated”
problem. If we assume that with each rebound j = 1, 2, 3, . . . the coefficient k j
changes, then
13
Fig. 1.6 Graph of the position of the bouncing ball above the floor
After a rebound, the ball gains the speed v + = k · v − , where v + is the speed at
which the new flight begins, v − is the speed at which the ball touched the surface,
and k is a coefficient depending on the properties of the rebound. With an absolutely
elastic rebound, k = 1, with an absolutely inelastic k = 0, with an elastic 0 < k < 1.
We are interested in the moments of time at which the bounces occur and the law of
change in the height of the ball above the floor, as shown in Fig. 1.6.
When using the model of continuous and discrete time, this problem can be solved
only as a sequence of individual tasks. The processes occurring at the time of the
rebound can be neglected, that is, assume that the rebound occurs instantly. With this,
the rebound phenomenon itself is replaced by the instantaneous action v + = k · v − .
The first formulation of the problem: continuous time model.
In this simple problem, the solution of equations for each individual flight (except
the first) can be obtained in the explicit form:
h(t) = v 0 · t −
g·t
2
2
,
v(t) = v 0 − g · t,
where v 0 is the ball speed after rebound. Then, the next bounce time T can be found
from:
h(T ) = 0 ⇒ T =
2 · v 0
g
.
Consistently solving the equations of flight after the next bounce, we obtain the
required points in time.
The second formulation of the problem: discrete time model.
We use the solution obtained in order to solve a slightly more “complicated”
problem. If we assume that with each rebound j = 1, 2, 3, . . . the coefficient k j
changes, then