A garden hose attached with a nozzle is used to fill a 10 L bucket. The inner diameter of the hose is 2 cm, and it reduces to 0.8 cm at the nozzle exit. If it takes 50 s to fill the bucket with water, determine the velocity of water at the nozzle exit. Express your answer in m/s.

Step 1: Determine the volume flow rate (Q) of the water. This is the volume of water that flows per unit time. Since the bucket has a volume of 10 L and it takes 50 s to fill, the volume flow rate is:

Q = Volume / Time = 10 L / 50 s = 0.2 L/s

Step 2: Convert the volume flow rate from L/s to m³/s. There are 1000 L in 1 m³, so:

Q = 0.2 L/s * (1 m³ / 1000 L) = 0.0002 m³/s

Step 3: The volume flow rate is also equal to the cross-sectional area of the nozzle exit (A) times the velocity of the water at the nozzle exit (v). So:

Q = Av

We can rearrange this equation to solve for v:

v = Q / A

Step 4: Determine the cross-sectional area of the nozzle exit. The area of a circle is given by the formula A = πr², where r is the radius. The radius is half the diameter, so r = 0.8 cm / 2 = 0.4 cm. Convert this to meters:

r = 0.4 cm * (1 m / 100 cm) = 0.004 m

Then calculate the area:

A = π * (0.004 m)² = 0.00005027 m²

Step 5: Substitute the values of Q and A into the equation for v:

v = 0.0002 m³/s / 0.00005027 m² = 3.98 m/s

So, the velocity of water at the nozzle exit is approximately 3.98 m/s.