Our model consists of a Brownian particle $X$ moving in $\mathbb{R}$, where a Poissonian field of moving traps is present. Each trap is a ball with constant radius, centered at a trap point, and each trap point moves under a Brownian motion independently of others and of the motion of $X$. Here, we investigate the 'speed' of $X$ on the time interval $[0,t]$ and on 'microscopic' time scales given that $X$ avoids the trap field up to time $t$. Firstly, following the earlier work of Athreya et al. [Math. Phys. Anal. Geom. 20:1 (2017)], we obtain bounds on the maximal displacement of $X$ from the origin. Our upper bound is an improvement of the corresponding bound therein. Then, we prove a result showing how the speed on microscopic time scales affect the overall macroscopic subdiffusivity on $[0,t]$. Finally, we show that $X$ moves subdiffusively even on certain microscopic time scales, in the bulk of $[0,t]$. The results are stated so that each gives an 'optimal survival strategy' for the system. We conclude by giving several related open problems.