Imagine an object falling along a quarter circle of height h meters (h would also be the radius) and gravity g meters per second squared. How long will it take for the object to travel the distance of the curve in terms of h and g? (no friction, no rolling, the object is negligibly small compared to the quarter circle, and the object is stationary until t = 0)

I was able to determine the following answer:

$${t(h, g)=\sqrt{\frac{h}{g}}\cdot1.854074...}$$

The only issue is that I have no idea where the magic 1.854074... number comes from. This was solved with brute force calculation and curve fitting, but I'm sure a proper solution to this problem will better explain the value.

Imagine an object falling along a quarter circle of height $h$ meters ($h$ would also be the radius) and gravity $g$ meters per second squared. How long will it take for the object to travel the distance of the curve in terms of $h$ and $g$? (no friction, no rolling, the object is negligibly small compared to the quarter circle, and the object is stationary until $t = 0$.)

I was able to determine the following answer:

$${t(h, g)=\sqrt{\frac{h}{g}}\cdot1.854074...}$$

The only issue is that I have no idea where the magic 1.854074... number comes from. This was solved with brute force calculation and curve fitting, but I'm sure a proper solution to this problem will better explain the value.