Poisson broken lines process and its application to Bernoulli first passage percolation

In this note we introduce a process, which we call 'the Poisson broken line s process', and we compute the intensity of a point process which is obtain ed by intersecting the Poisson broken lines process with an abscissa axis. In the second part we apply this result to compute an explicit lower bound for the time constant of a planar Bernoulli first passage percolation model with the parameter p < p(c).