Alright! Let's walk through the steps to solve this problem.
Given:
- Power of the pump = 2 kW
- Height = 10 meters
- Power needs to be converted to watts: 1 kW = 1000 W, so 2 kW = 2000 W
We need to find out how much water (mass) the pump can lift every minute to a height of 10 meters.
Firstly, recall that power is the rate at which work is done. The formula for power in terms of lifting a mass at a height is:
[tex]\[ \text{Power} = \frac{\text{Work}}{\text{Time}} \][/tex]
And the work done against gravity to lift a mass [tex]\(m\)[/tex] to a height [tex]\(h\)[/tex] is given by:
[tex]\[ \text{Work} = m \cdot g \cdot h \][/tex]
where:
- [tex]\(m\)[/tex] is the mass
- [tex]\(g\)[/tex] is the acceleration due to gravity (approximately [tex]\(9.8 \, \text{m/s}^2\)[/tex])
- [tex]\(h\)[/tex] is the height
Thus,
[tex]\[ \text{Power} = \frac{m \cdot g \cdot h}{\text{Time}} \][/tex]
Given that the pump's power is [tex]\(2000 \, \text{W}\)[/tex], and wanting to calculate the mass lifted per second means we set the time to 1 second:
[tex]\[ 2000 = m \cdot 9.8 \cdot 10 \][/tex]
Now solve for [tex]\(m\)[/tex]:
[tex]\[ m = \frac{2000}{9.8 \cdot 10} \][/tex]
The value of [tex]\(m\)[/tex] turns out to be approximately [tex]\(20.408 \, \text{kg}\)[/tex].
This is the mass of water the pump can lift every second.
Now, we need to determine how much water the pump will lift every minute. Since there are 60 seconds in a minute:
[tex]\[ \text{Mass per minute} = 20.408 \times 60 \][/tex]
The mass of water that the pump will lift every minute is approximately [tex]\(1224.49 \, \text{kg}\)[/tex].
Therefore, the electric pump can lift about [tex]\(1224.5 \, \text{kg}\)[/tex] of water every minute to a height of 10 meters.
