  In this paper we derive, directly from the Nambu-Goto action, the relevant
components of the acceleration of cosmological featureless $p$-branes,
extending previous analysis based on the field theory equations in the
thin-brane limit. The component of the acceleration parallel to the velocity is
at the core of the velocity-dependent one-scale model for the evolution of
$p$-brane networks. We use this model to show that, in a decelerating expanding
universe in which the $p$-branes are relevant cosmologically, interactions
cannot lead to frustration, except for fine-tuned non-relativistic networks
with a dimensionless curvature parameter $k \ll 1$. We discuss the implications
of our findings for the cosmological evolution of $p$-brane networks.
