  We prove that front tracking approximations to entropy solutions of scalar
conservation laws with convex fluxes converge at a rate of $\Delta x^2$ in the
1-Wasserstein distance $W_1$. Assuming positive initial data, we also show that
the approximations converge at a rate of $\Delta x$ in the $\infty$-Wasserstein
distance $W_\infty$. Moreover, from a simple interpolation inequality between
$W_1$ and $W_\infty$ we obtain convergence rates in all the $p$-Wasserstein
distances: $\Delta x^{1+1/p}$, $p \in [1,\infty]$.
