About simple moment Today I encountered a quite weird problem.
There are two situations in the picture:

The first situation: The cart move in a frictionless straight line from A to B
The second situation: The cart move in a frictionless line from A to B, but
there is a slightly curved hole on that line.
The question of the problem is which cart arrive at B faster.
The answer is the second cart reach B before the first cart. My teacher explanation is: When moving down the hole, the x-coordinate speed of the second cart was increased. When the cart reach the other side of the hole, the speed return to normal. Therefore, the time the second cart travel from A to B ( AB only related to x-coordinate) is shortened.
I wondered whether my teacher was right? and if he was correct why don't we apply "the hole" in transport that need speed like bullet train?
 A: Your teacher is right. To make the answer more intuitive, one can note that, in Fig. 2, it takes the cart the same time to go from the top at A to the bottom as from the bottom to the top at B, and, compared to Fig. 1, there is a positive component of force along x (along AB) when the cart goes from the top at A to the bottom.     
A: The cart can gain speed by rolling downhill, and benefit of that additional speed can exceed the cost of needing traveling a greater total distance than the straight line.
The general name for this question is the Brachistochrone Problem, and the general shape of the optimal solution is a portion of a cycloid.
A: This can be understood also as follows: Assume the initial horizontal speed is infinitesimal small and (A) and (B) are at the corners of the curved track. 
In case 1) the car needs an infinitely long time to travel from A to B. In case 2) if starting at the kink, the car will have reached the lowest point (depth $h$) in a finite time $T$ and will have there a speed of $\sqrt{2gh}$. 
Thus, it reaches (if there is no friction) B at a finite time $2T$ which is in contrast to the first case.
