Final answer:
To determine the speed of water in the narrow section of the pipe, we apply the continuity equation, which leads to the conclusion that water will flow 16 times faster due to the reduction of the pipe diameter to one-fourth of its original size, resulting in a speed of 16.00 m/s.
Step-by-step explanation:
The question inquires about the change in speed of water flow as it moves from a wider section of a pipe to a narrower section. To solve the problem, we'll utilize the principle of conservation of mass, which is also known as the continuity equation in fluid dynamics. This principle states that, for an incompressible fluid, the product of the cross-sectional area (A) and the velocity (v) of the fluid remains constant throughout its flow in a closed system.
The continuity equation is expressed as A₁ * v₁ = A₂ * v₂, where A₁ and v₁ are the area and velocity of the fluid in the first section of the pipe, and A₂ and v₂ are the respective values in the second section. Given that the pipe narrows to one-fourth its original diameter, we can deduce that the cross-sectional area of the pipe is reduced to one-sixteenth (since the area is proportional to the square of the diameter), and therefore the speed at which the water flows through the narrow section increases by a factor of sixteen to maintain the flow rate.
Using the given information, v₂ = (A₁/A₂) * v₁. Since A₁/A₂ is the square of the diameter ratio, it is 16, and we have the initial speed v1 = 1.00 m/s, thus calculating the new speed v₂ = 16 * 1.00 m/s = 16.00 m/s.