Prey rely on making correct risk assessments when approached by potential predators in order to stay alive. We conducted experimental human approaches with different simulated threat levels toward solitary adult male Thomson’s gazelles, that were located in open grassland. We measured individuals flight initiation distance (FID), distance fled, escape speed, and the distance between the location where the focal individual had stopped to flee and where the human stopped the approach (termed safety distance). Multivariate analyses revealed an overall significant effect of starting distance, alertness, and time of day, but no statistical effect was found for approach speed on the multivariate response. The individual responses showed a significant positive effect of starting distance on both FID and safety distance. We also found a novel unimodal effect of time on FID. Finally, alertness and approach speed only had a significant effect on safety distance, where faster approaches and individuals that displayed alert behavior had shorter safety distances. Together, these findings indicate support for the “flush early and avoid the rush” hypothesis, shows the necessity of using starting distance, alertness, and time as covariates when testing the effects of threat level, and demonstrate the usefulness of the new metric safety distance.