Abstract Using the factorized form of the leading behavior (in powers of Q) of fixed-angle elastic hadron-hadron scattering given by Botts and Sterman, the leading perturbative QCD contribution to dσ/dt (ϑ = π/2) for pions and protons is numerically evaluated. Various models of the long-distance, nonperturbative hadronic wave functions are used and the different soft cutoff prescriptions made explicit. For the proton case, it is found that the size of the Landshoff contribution is of the same order as the observed data, though in that energy range the resulting cross sections are strongly dependent on momenta cutoffs, choice of wave function and the small ξ contribution. The onset of asymptopia, defined to be where soft effects become negligible, is found to occur at ln(s/1 GeV2) ≈ 4 for pions and ln(s/1 GeV2) ≈ 8 for protons. Logarithmic effects swamp out the power-law predictions of Botts and Sterman and Mueller, dσ/dt|ππαs−5.83 and dσ/dt|ppα s−9.59, until ln(s/1 GeV2) ≈ 20.