Exact solution of the one-dimensional ballistic aggregation

An exact expression for the mass distribution $\rho (M,t)$ of the ballistic aggregation model in one dimension is derived in the long time regime. It is shown that it obeys scaling $\rho (M,t)=t^{-4\mathrm{/}3}F(M\mathrm{/}{t}^{2\mathrm{/}3})$ with a scaling function $F(z)\sim z^{-1\mathrm{/}2}$ for $z\ll 1$ and $F(z)\sim \exp (-z^3\mathrm{/}12)$ for $z\gg 1$. Relevance of these results to Burgers turbulence is discussed.