Water flows with constant speed through a garden hose that goes up a step 20.0 cm high.
1. If the water pressure is 143 kPa at the bottom of the step, what is its pressure at the top of the step?
Now suppose the cross sectional area of the hose at the top of the step is reduced to half that at the bottom of the step and that the speed of the water at the bottom of the step is 1.20 m/s.
2. What is the speed of the water at the top of the step?
3. What is the pressure at the top of the step?
To answer these questions, we can use Bernoulli's equation, which states that the total pressure at a given point in a flowing fluid system is equal to the sum of the kinetic pressure (related to the speed of the fluid) and the potential pressure (related to the height of the fluid).
1. To find the pressure at the top of the step, we can assume that the water flow is ideal and that there is no energy loss due to friction. In this case, since the speed of the water is constant throughout the hose, the kinetic pressure remains the same. Therefore, the pressure at the top of the step is equal to the pressure at the bottom of the step.
So, the pressure at the top of the step is also 143 kPa.
2. Now, let's consider the situation where the cross-sectional area of the hose at the top of the step is reduced to half that at the bottom, and the speed of the water at the bottom of the step is 1.20 m/s. We can find the speed of the water at the top of the step using the principle of continuity.
The principle of continuity states that the product of the cross-sectional area and the fluid velocity at any point in a closed system is constant. Therefore, we can write:
A1 * v1 = A2 * v2
where A1 and v1 are the cross-sectional area and velocity at the bottom of the step, and A2 and v2 are the cross-sectional area and velocity at the top of the step.
In this case, we know that A2 = (1/2) * A1 (since the area is reduced to half). Substituting this into the equation, we have:
A1 * v1 = (1/2) * A1 * v2
Simplifying, we find:
v2 = 2 * v1
So, the speed of the water at the top of the step is 2 * 1.20 m/s, which is 2.40 m/s.
3. To find the pressure at the top of the step, we can apply Bernoulli's equation again. Since the speed of the water has increased, the kinetic pressure at the top of the step will be greater than at the bottom. However, the potential pressure remains the same, as the height of the fluid does not change.
Therefore, the pressure at the top of the step can be calculated as follows:
P2 = P1 + (1/2) * ρ * (v1^2 - v2^2)
where P1 is the pressure at the bottom of the step, P2 is the pressure at the top of the step, ρ is the density of water (which is approximately 1000 kg/m^3), v1 is the speed of the water at the bottom of the step (1.20 m/s), and v2 is the speed of the water at the top of the step (2.40 m/s).
Substituting the given values, we can calculate the pressure at the top of the step.