Sol.

When sides AB and DC of the rhombus (fig.) are almost horizontal, it is at once obvious that the ball will roll faster down sides AD and DC (the second case in the problem). This can be seen from the fact that the ball will travel along DC at a high average velocity, acquired from its motion alongside AD. But in the first the case ball will travel along AB with a very small average velocity (since its acceleration is small). The result found for this particular case remains true for the general case, as may be verified from the following calculation. Let sides AB and DC from an angle of α with the horizontal and sides BC and AD an angle of β with the vertical. If the ball rolls along sides AB and BC it spends time t 1 + t 2 on this, where t 1 is spent on traveling along AB and t 2 on traveling along BC. The acceleration during motion along AB equals g sin α . Therefore, to calculate t 1 we have the equation

A =

The acceleration during motion along BC equals g cos β , and this motion takes place with an initial velocity of , so we can find t 2 by solving the equation

A =

For the second case we shall have the same expressions with the only difference that the acceleration g sin α and and g cos β must change places. So the sum of the two times for the first case will be

+ ,

And for the second case the sum of the two times will be

+

Since it is clear that g sin α is less than g cos β , we find that the sum of the two times in the first case is greater than in the second.