Your town is installing a fountain in the main square. If the water is to rise 14 m (45.9 feet) above the fountain, how much pressure must the water have as it moves slowly toward the nozzle that sprays it up into the air? Assume atmospheric pressure equal to 100,000 Pa.

Question

Your town is installing a fountain in the main square. If the water is to rise 14 m (45.9 feet) above the fountain, how much pressure must the water have as it moves slowly toward the nozzle that sprays it up into the air? Assume atmospheric pressure equal to 100,000 Pa.

Answer 1

To find the pressure needed for the water to rise 14 m above the fountain, we can make use of the concept of hydrostatic pressure.

Hydrostatic pressure is given by the equation:

P = ρgh

where P is the pressure, ρ is the density of the fluid, g is the acceleration due to gravity, and h is the height of the fluid column.

In this case, the height of the fluid column is 14 m. The density of water is approximately 1000 kg/m^3, and the acceleration due to gravity is 9.8 m/s^2.

Given these values, we can now calculate the pressure.

P = (1000 kg/m^3) * (9.8 m/s^2) * (14 m)
P = 137,200 Pa

So, the water must have a pressure of approximately 137,200 Pa as it moves slowly toward the nozzle that sprays it up into the air.

Note: We assumed that the atmospheric pressure is equal to 100,000 Pa, but this does not affect the pressure required for the water to rise 14 m above the fountain. This value is only relevant when considering the difference in pressure between the water and the atmosphere.

Answer 2

To calculate the pressure required for the water to rise 14 meters above the fountain, we can use the concept of pressure due to height of a fluid. The formula for calculating pressure due to height is:

Pressure = density * gravitational acceleration * height

Here, we assume the density of water is 1000 kg/m³ and the gravitational acceleration is approximately 9.8 m/s².

Step 1: Convert the height from feet to meters (since the formula requires height in meters).
14 meters * 3.28 feet/meter = 45.92 feet

So, the height is approximately 45.92 meters.

Step 2: Calculate the pressure due to the height.
Pressure = (density of water) * (gravitational acceleration) * (height)
Pressure = 1000 kg/m³ * 9.8 m/s² * 45.92 m

Using a calculator, we find:
Pressure = 447,136 Pa

Step 3: Add the atmospheric pressure.
Total Pressure = Pressure due to height + Atmospheric pressure
Total Pressure = 447,136 Pa + 100,000 Pa

Using a calculator again, we find:
Total Pressure = 547,136 Pa

Therefore, the water must have a pressure of approximately 547,136 Pa as it moves slowly toward the nozzle that sprays it up into the air.