Paul Ekman suggests that in near-accidents, fear motivates people to hit their car brakes. Is the fear response fast enough to do this?
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.