Sure! Let's go through the problem step-by-step to determine how much water the pump will lift every minute to a height of 10 meters.
1. Identify the given information:
- Power of the pump: 2 kW (kilowatts)
- Height to which the water is lifted: 10 meters
- Time duration: 1 minute
2. Convert units where necessary:
- Power in watts: Since 1 kW is equal to 1000 watts, the power of the pump in watts is:
[tex]\(2 \, \text{kW} = 2 \times 1000 \, \text{W} = 2000 \, \text{W}\)[/tex]
- Time in seconds: Since 1 minute is equal to 60 seconds, the time duration in seconds is:
[tex]\(1 \, \text{minute} = 1 \times 60 \, \text{s} = 60 \, \text{s}\)[/tex]
3. Understand the relationship between power, work, and energy:
- The power of the pump indicates the rate at which work is done. The work done in lifting the water is equal to the gravitational potential energy gained by the water.
- Gravitational potential energy (work done) is given by [tex]\( \text{Work done (J)} = \text{Mass (kg)} \times \text{Gravity (m/s}^2\text{)} \times \text{Height (m)} \)[/tex]
4. Relate power to work done and time:
- Power is defined as the work done per unit time: [tex]\( \text{Power (W)} = \frac{\text{Work done (J)}}{\text{Time (s)}} \)[/tex]
- Rearranging this formula to find work done: [tex]\( \text{Work done (J)} = \text{Power (W)} \times \text{Time (s)} \)[/tex]
5. Find the work done by the pump in 60 seconds:
- [tex]\( \text{Work done (J)} = 2000 \, \text{W} \times 60 \, \text{s} = 120,000 \, \text{J} \)[/tex]
6. Set up the equation to find the mass of water lifted:
- We know the work done (120,000 J) and the height (10 meters) and we are using the gravitational acceleration constant [tex]\( \text{g} = 9.81 \, \text{m/s}^2 \)[/tex]
- Rearrange the gravitational potential energy formula to solve for mass:
[tex]\( \text{Mass (kg)} = \frac{\text{Work done (J)}}{\text{Gravity (m/s}^2\text{)} \times \text{Height (m)}} \)[/tex]
7. Substitute the known values into the formula:
- [tex]\( \text{Mass (kg)} = \frac{120,000 \, \text{J}}{9.81 \, \text{m/s}^2 \times 10 \, \text{m}} \approx 1223.24 \, \text{kg} \)[/tex]
Therefore, the electric pump will lift approximately 1223.24 kg of water every minute to a height of 10 meters.
