A rain cloud contains 2.66 x 10^7 kg of water vapor. How long will it take for a 2.00 kW pump to raise the same amount of water to the cloud's altitude, 2.00 km?
To find out how long it will take for the pump to raise the same amount of water to the cloud's altitude, we need to calculate the work required and then use the power of the pump to determine the time.
First, let's calculate the work required to lift the water to the cloud's altitude. The work done is given by the formula:
Work = force x distance
In this case, the force required to lift the water is equal to the weight of the water. The weight of an object is given by the formula:
Weight = mass x gravity
The mass of the water vapor is given as 2.66 x 10^7 kg. The acceleration due to gravity is approximately 9.8 m/s^2. Therefore, the weight of the water vapor is:
Weight = (2.66 x 10^7 kg) x (9.8 m/s^2)
Next, we need to calculate the distance over which the pump needs to raise the water. The altitude of the cloud is given as 2.00 km, which is equal to 2000 m.
Now, we can calculate the work required:
Work = Weight x distance
Once we have the work required, we can use the power of the pump to determine the time. Power is the rate at which work is done and is given by the formula:
Power = work / time
Rearranging the formula, we can solve for time:
Time = work / power
Now, let's plug in the values and calculate the time.
To find the time it will take for the 2.00 kW pump to raise the amount of water to the cloud's altitude, we need to determine the amount of work required and then calculate the time using the power formula.
Given:
Mass of water vapor in the cloud = 2.66 x 10^7 kg
Power of the pump = 2.00 kW
Height of the cloud's altitude = 2.00 km
Step 1: Convert the power from kilowatts (kW) to watts (W).
1 kW = 1000 W
So, the power of the pump in watts is: 2.00 kW x 1000 = 2000 W.
Step 2: Calculate the work required to raise the water to the cloud's altitude.
Work (W) = Force (F) x Distance (d)
In this case, the force required to raise the water is equal to the weight, which is given by:
Force = mass x acceleration due to gravity (g)
The acceleration due to gravity is approximately 9.8 m/s^2.
Force = 2.66 x 10^7 kg x 9.8 m/s^2 = 2.6128 x 10^8 N
The distance (altitude) is given as 2.00 km, which is equal to 2.00 x 10^3 m.
Work = Force x Distance = 2.6128 x 10^8 N x 2.00 x 10^3 m = 5.2256 x 10^11 J
Step 3: Calculate the time using the power formula.
Power (P) = Work (W) / Time (t)
Rearranging the formula, we can solve for time:
Time (t) = Work (W) / Power (P)
Time (t) = 5.2256 x 10^11 J / 2000 W = 2.6128 x 10^8 seconds
Therefore, it will take approximately 2.6128 x 10^8 seconds for the 2.00 kW pump to raise the same amount of water to the cloud's altitude of 2.00 km.
time=work/power=massrain*g*height/power
change power to w, distance to m, solve for time in seconds.