Will free-fall object into black hole exceed speed of light $c$ before hitting black hole surface? In Newtonian mechanics, if we throw an object in against direction of gravity with speed $v$ and it achieve max height of $h$. Now if we allow object to fall from that height $h$, it will eventually attain speed $v$ when it reach position where we launch it. 
Now applying same idea to a black hole in general relativity. Speed require to escape black hole gravity is greater than $c$, so if we throw something into black hole with almost the speed of light, the object speed will exceed speed of light $c$ before hitting black hole surface! How relativity explain this? Can space-time curvature reduce speed of this freely falling object from attaining speed of light?
 A: Nope, it will just fall in a reasonable amount of time (if you go with it, but watch out for tidal forces!), or take forever to fall in (if you are watching from outside).
Also, if I may be so bold as to suggest doing some quantum mechanics instead of kinematics while you are there, you could probably lock down some funding no problem.
A: To answer your question you need to be clear what coordinates you're using. If you use coordinates that are co-moving with the rock falling into the black hole then the rock will see the event horizon pass at the speed of light.
External observers, using Schwarzchild coordinates, will see the rock slow down as it approaches the horizon, and if you wait an infinite time you'll see it stop.
External observers obviously can't comment on the speed of the rock after it has passed the event horizon because it takes longer than an infinite time to get there. If you use the rock co-moving coordinates then you can ask what speed you hit the singularity and ... actually I'm not sure what the answer is. I'll have to go away and think about it.
Incidentally http://jila.colorado.edu/~ajsh/insidebh/schw.html is a fun site describing what happens when you fall into a black hole.
A: It will reach the speed of light exactly at the black hole surface.
A: 
if we throw something into black hole with almost the speed of light,
  the object speed will exceed speed of light c before hitting black
  hole surface! How relativity explain this? Can space-time curvature
  reduce speed of this freely falling object from attaining speed of
  light?

If you throw (or drop someting for that matter) radially inwards towards a black hole the gravitational acceleration in coordinate time will go as:
$$\frac{d\bar{v}}{dt}=-\frac{GM}{r^2}\hat{r}(1-2\frac{v^2}{c^2(1-\frac{2GM}{rc^2})}-\frac{v^2}{c^2(1-\frac{2GM}{rc^2})^2})$$
The two extra terms will prevent any object moving radially inwards  from reaching the speed of light. You can only reach the Schwarzschild radius of $r=2GM/c^2$ if you move infinitely slow. Usually you do not talk of the Schwarzshild radius as the "black hole surface", but I guess that is what you mean.
