Paul Ekman suggests that in near-accidents, fear motivates people to hit their car brakes. Is the fear response fast enough to do this? Recall a time when you were driving your car and suddenly
  another car appeared, going very fast, seeming as if it were about to
  hit you. Your conscious mind was focused on an interesting conversation with a friend in the passenger's seat or the program on the
  radio. In an instant, before you had time to think, before the conscious, self-aware part of your mind could consider the matter, danger was sensed and fear began.
  As an emotion begins, it takes us over in those first milliseconds,
  directing what we do and say and think. Without consciously choosing to do it, you automatically turned the steering wheel to avoid the
  other motorist, hitting the brake with your foot. 


Paul Ekman writes in "Emotions Revealed: Recognizing Faces and Feelings to Improve Communication and Emotional Life" that in near-accidents people hit the brakes because they feel fear.

Is the fear response fast enough to do that or do people first hit the brakes and then switch to an emotional state of fear?
This was much longer than I expected! There's quite a bit of ground to cover, but I try to go over it quickly.

So, there are two implicit theoretical assumptions in your question:


We have an "affect program" for fear in our brain (e.g., Ekman &amp; Cordaro, 2011).  When the fear program is activated, a specific pattern of changes in experience, behavior, physiology, and cognition occur (i.e., the fear response).
Present emotions are the primary causes of future behavior.


I'll briefly examine these assumptions, and then I'll draw a conclusion.

Assumption #1:  Affect programs

By "affect program" I mean a discrete module or circuit in the brain that, when activated, automatically produces a fear response.  So if I see a car about to hit me, it might activate my fear program, which will produce my fear response.

Your question uses this assumption by asking about the speed of the fear response.  How fast can my fear program activate my fear response (which would motivate me to hit the brakes)? Does my fear program trigger a response too slowly, such that my fear response will occur after I hit the brakes?

In recent years, the "affect program" idea has been called into question (e.g., Barrett, 2006; Lindquist et al., 2012).  As it stands, it's a fairly controversial assumption that I would argue has iffy empirical support and explanatory power.

If there isn't a "fear program," then what is there? Well…that's tricky. I could get into a whole lot of theory, but that would take up a lot of space (although if you're interested, see Satpute, Wilson-Mendenhall, &amp; Barrett, 2015).  I think the point that I want to make is that it's difficult to talk about the speed of the fear response.  Without having a consensual definition of "emotion," we can't talk very easily about the nature of fear (i.e., what it is, what it looks like, when it begins, when it ends, its function, its neural basis, its evolution, etc.).

Assumption #2: Present emotions cause future behavior

The idea that fear causes me to run away, that anger causes me to approach, or that sadness causes me to withdraw seems very intuitive.  In fact, despite its mixed empirical support, many researchers seem to accept this idea without reservation.

On the other hand, Roy Baumeister and colleagues have argued that present emotions affect future behaviors indirectly (Baumeister, Vohs, DeWall, &amp; Zhang, 2007). In fact, they argue that behavior tends to follow emotions, not the other way around.

To make a long story short, Baumeister argues that anticipated emotions are the more direct causes of present behavior.  So If I anticipate feeling relieved as a consequence of hitting the brakes, then I will hit the brakes.  That is, I will hit the brakes because this will lead to future relief. My fear is not motivating this behavior; instead, my expectation of relief is. Indeed, a recent small-scope meta-analysis supports this theory (DeWall, Baumeister, Chester, &amp; Bushman, 2015)

Conclusion

So I made things pretty complicated! I posed two questions:


Can we meaningfully talk about the speed of the fear response without a consensual definition of emotion?
Do present emotions cause future behavior (e.g., fear causing me to hit the brakes)?


My answers to both questions were no.  However, I think if we ignore the yuckiness of the first question, we can say that fear probably doesn't cause us to hit the brakes, but rather the expectation of relief does (or some other anticipated feeling).   

When and how quickly the fear response occurs during this situation is hard to answer (but can be resolved somewhat depending on your theoretical perspective). If we assume that emotions are multi-componential, then we might imagine that the experience component need not occur at the same moment as the behavioral component (mirroring studies of emotion coherence, e.g., Mauss et al., 2005).  Thus, the experience of fear could (hypothetically) occur before or after the braking behavior.