  Let $L_n$ be a line graph with $n$ vertices and let $I$ be its $t$-path
ideal. It is shown that $I^s$ has a linear resolution for some $s\geq 1$ (or
equivalently for all $s\geq 1$) if and only if $I^s$ has linear quotients for
some $s\geq 1$ (or equivalently for all $s\geq 1$) if and only if $t\leq n\leq
2t$. In addition, we present an explicit formula for the regularity of $I^s$
for all $s\geq 1$. It turns out it is linear in $s$ from the very beginning.
