A particle falls along a cycloidal path from the origin to the final point \((x, y)=\) \((\pi a / 2, a)\); the time required is \(\pi \sqrt{a / 2 g}\). How long would it take the particle to slide along a straight-line path between the same points? Express the time for the straight-line path in the form \(t_{\text {straight }}=k t_{\text {cycloid }}\), and find the numerical factor \(k\).