  We use $(4+1)$ split to derive the 4D induced energy density $\rho $ and
pressure $p$ of the $5D$ Ricci-flat cosmological solutions which are
characterized by having a bounce instead of a bang. The solutions contain two
arbitrary functions of time $t$ and, therefore, are mathematically rich in
giving various cosmological models. By using four known energy conditions
(null, weak, strong, and dominant) to pick out and study physically meaningful
solutions, we find that the 4D part of the $5D$ solutions asymptotically
approach to the standard 4D FRW models and the expansion of the universe is
decelerating for normal induced matter for which all the four energy conditions
are satisfied. We also find that quintessence might be normal or abnormal,
depending on the parameter $w$ of the equation of state. If $-1\leqslant
w<-1/3$, the expansion of the universe is accelerating and the quintessence is
abnormal because the strong energy condition is violated while other three are
satisfied. For phantom, all the four energy conditions are violated. Before the
bounce all the four energy conditions are violated, implying that the cosmic
matter before the bounce could be explained as a phantom which has a large
negative pressure and makes the universe bouncing. In the early times after the
bounce, the dominant energy condition is violated while the other three are
satisfied, and so the cosmic matter could be explained as a super-luminal
acoustic matter.
