Detecting a moderately-fast moving object from 1ft away
As far as the speed, IR detectors are definitely fast enough with response time being of the order of ns or us (if you use more filtering). 
There are a couple of areas of concern:

1) The pattern is probably close to being omni-directional (I could not see the datasheet that would show this) so you may have issues with the signal bouncing off of the floor, other objects and would have to experiment a lot with thresholds.

2) It is likely that the signal would not increase/decrease monotonically as the car is passing by, so you would have to implement some sort of a debounce function to avoid multiple counts for the same event.