A stream of water flows vertically downwards from a running tap, as shown below. Some way down the flow, there is a 3 cm long segment of flowing water where the diameter of the circular stream reduces from d₁ = 5 mm to a diameter d₂ = 4 mm. From this we can determine the flow rate and how long it will take to fill a beaker of volume 200 cm³. We shall assume that water is incompressible.
[Figure: Diagram showing water stream narrowing from 5 mm diameter to 4 mm diameter over 3 cm length]
a) Explain why the segment of water becomes narrower.
b) If the speed of the water at the top of the segment is v₁ then what is the speed v₂ of the water at the bottom of the segment expressed in terms of v₁, d₁ and d₂?
c) Calculate the speed of the water flow at the top of the segment. You may want to use the equation of motion v² - u² = 2as
d) From your answer to part (b), calculate the volume flow of water per second.
e) Calculate the time taken to fill a 200 cm³ beaker.