To find the power of the motor lifting and delivering water, we need to calculate the work done in two parts: lifting the water from the well and delivering it with a certain velocity. Let's break this down step by step.
### Step 1: Calculate the Work Done in Lifting the Water
The work done in lifting the water from the well can be calculated using the formula:
\[
\text{Work} = m \cdot g \cdot h
\]
Where:
- \( m \) = mass of water (in kg)
- \( g \) = acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \))
- \( h \) = height (in meters)
Given:
- The volume of water lifted per second = 10 liters = 10 kg (since 1 liter of water has a mass of approximately 1 kg)
- Height \( h = 20 \, \text{m} \)
Substituting the values:
\[
\text{Work}_{\text{lifting}} = 10 \, \text{kg} \cdot 9.81 \, \text{m/s}^2 \cdot 20 \, \text{m}
\]
\[
\text{Work}_{\text{lifting}} = 1962 \, \text{J} \, (\text{Joules})
\]
### Step 2: Calculate the Work Done in Delivering the Water
The work done in delivering the water at a velocity can be calculated using the kinetic energy formula:
\[
\text{Work} = \frac{1}{2} m v^2
\]
Where:
- \( v \) = velocity (in m/s)
Given:
- Velocity \( v = 10 \, \text{m/s} \)
Substituting the values:
\[
\text{Work}_{\text{delivery}} = \frac{1}{2} \cdot 10 \, \text{kg} \cdot (10 \, \text{m/s})^2
\]
\[
\text{Work}_{\text{delivery}} = \frac{1}{2} \cdot 10 \cdot 100
\]
\[
\text{Work}_{\text{delivery}} = 500 \, \text{J} \, (\text{Joules})
\]
### Step 3: Calculate the Total Work Done
Now, we can find the total work done by adding the work done in lifting and delivering:
\[
\text{Total Work} = \text{Work}_{\text{lifting}} + \text{Work}_{\text{delivery}}
\]
\[
\text{Total Work} = 1962 \, \text{J} + 500 \, \text{J}
\]
\[
\text{Total Work} = 2462 \, \text{J}
\]
### Step 4: Calculate the Power of the Motor
Power is defined as work done per unit time. Since the work is done in one second:
\[
\text{Power} = \frac{\text{Total Work}}{\text{Time}}
\]
\[
\text{Power} = \frac{2462 \, \text{J}}{1 \, \text{s}}
\]
\[
\text{Power} = 2462 \, \text{W}
\]
To convert this into kilowatts:
\[
\text{Power} = \frac{2462}{1000} = 2.462 \, \text{kW}
\]
### Final Answer
The power of the motor is approximately **2.46 kW**.
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