Final answer:
A bucket fills faster at the lower stairs than at the upper stairs due to the principles of fluid dynamics. Specifically, the continuity equation dictates that as water flows downwards, its velocity increases with the decrease in cross-sectional area, meaning that the water hits the bucket with greater speed and therefore fills it quicker.
Step-by-step explanation:
You've asked why a bucket is filled faster at the lower stairs than at the upper stairs. This phenomenon can be explained by principles in physics, specifically fluid dynamics. According to the continuity equation, for incompressible fluids, the product of cross-sectional area (A) and velocity (V) remains constant throughout the flow. If a fluid flows down the stairs (like water in a waterfall or fountain), as the water descends, the cross-sectional area of the stream decreases. Because AV is constant, if the area decreases, the velocity increases.
This is similar to the way water flows faster as a faucet stream narrows because gravity is pulling the water downwards, increasing its speed and causing the stream to narrow to conserve momentum. This is the principle that the velocity times the cross-sectional area entering a region must equal the velocity times the cross-sectional area leaving that region, assuming the density remains constant. From this, we can deduce that the bucket fills faster at the lower stairs because the water flows with a higher velocity due to gravitational acceleration and conservation of momentum within a decreased cross-sectional area.