As a significant fraction of stars are in multiple systems, binaries play a crucial role in stellar evolution. Among short-period (<1 day) binary characteristics, age remains one of the most difficult to measure. In this paper, we constrain the lifetime of short-period binaries through their kinematics. With the kinematic information from Gaia Data Release 2 and light curves from {it Wide-field Infrared Survey Explorer} (WISE), we investigate the eclipsing binary fraction as a function of kinematics for a volume-limited main-sequence sample. We find that the eclipsing binary fraction peaks at a tangential velocity of $10^{1.3-1.6}$ km/s, and decreases towards both low and high velocity end. This implies that thick disk and halo stars have eclipsing binary fraction $gtrsim 10$ times smaller than the thin-disk stars. This is further supported by the dependence of eclipsing binary fraction on the Galactic latitude. Using Galactic models, we show that our results are inconsistent with any known dependence of binary fraction on metallicity. Instead, our best-fit models suggest that the formation of these short-period binaries is delayed by 0.6-3 Gyr, and the disappearing time is less than the age of the thick disk. The delayed formation time of $gtrsim0.6$ Gyr implies that these short-period main-sequence binaries cannot be formed by pre-main sequence interaction and the Kozai-Lidov mechanism alone, and suggests that magnetic braking plays a key role in their formation. Because the main-sequence lifetime of our sample is longer than 14 Gyr, if the disappearance of short-period binaries in the old population is due to their finite lifetime, our results imply that most ($gtrsim90$%) short-period binaries in our sample merge during their main-sequence stage.
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