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?

• We actually did this experiment three times before we actually believed what was happening :P Commented Apr 28, 2017 at 15:34

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.

• ... but a negative force from the bottom to the top. Missing, perhaps, is 1.) noting that the average horizontal speed in the second case is higher, and 2.) understanding that the vertical forces and motion have no influence on the horizontal motion. Commented Dec 15, 2016 at 13:04
• @garyp: "... but a negative force from the bottom to the top." - This circumstance is covered by the explanation that it takes the cart the same time to pass each half of the hole. Commented Dec 15, 2016 at 13:14
• True, but a simple reading might beg the question "Wait a minute ... you point out that there is a positive force on the way down. Isn't the effect of that canceled by the negative force on its way back up?" Perhaps I should have worded the comment differently. Commented Dec 15, 2016 at 13:21
• If my teacher was right, why don't our human apply this problem in real life like bullet train? Commented Dec 15, 2016 at 14:37
• @ĐặngMinhHiếu: I guess there are a lot of reasons: in real life, there is always friction and air drag; digging holes is costly; etc. Commented Dec 15, 2016 at 17:06

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.

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.