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?
m=26600000kg
P=2000W
d=2000m
t=?
P=W/t=Fd/t
Forces that act on the object are Force that the pump applies (upward) and Fg (downward).
Fnet = F - Fg = 0, so F=Fg=26600000 *9.81
t=Fd/P=2.61*10^8 s
timeinseconds=masswater*g*height/power
Why did the rain cloud become friends with the pump? Because they both needed a little "altitude" adjustment!
But on a more serious note, let's do some calculations. To find the time it takes for the pump to raise the same amount of water, we can use the power equation:
Power (P) = Work (W) / Time (t)
Given that the power of the pump is 2.00 kW, we need to convert it to watts:
2.00 kW = 2.00 x 10^3 W
Now let's calculate the work done by the pump. Work is equal to the force applied multiplied by the distance covered:
Work (W) = Force (F) x Distance (d)
In this case, the force required to raise the water is equal to its weight. The weight of an object is given by:
Weight (W) = Mass (m) x Gravity (g)
Given that the mass of the water is 2.66 x 10^7 kg and the acceleration due to gravity is approximately 9.8 m/s^2, we can calculate the weight of the water:
Weight (W) = 2.66 x 10^7 kg x 9.8 m/s^2
Finally, we need to calculate the distance covered by the water, which is the cloud's altitude of 2.00 km. To convert it to meters:
Distance (d) = 2.00 km x 1000 m/km
Now we have all the information we need to solve for time. Let's plug the values into the equation:
2.00 x 10^3 W = (2.66 x 10^7 kg x 9.8 m/s^2) x (2.00 km x 1000 m/km) / t
Now we can solve for t.
To find the time it will take for the pump to raise the same amount of water to the cloud's altitude, we'll need to calculate the work done by the pump.
The work done by the pump (W) is equal to the power (P) multiplied by the time (t):
W = P * t
First, let's convert the power from kilowatts (kW) to watts (W):
1 kW = 1000 W
So, the power (P) is:
P = 2.00 kW = 2.00 * 1000 W = 2000 W
Now, let's calculate the work done by the pump.
The work (W) is equal to the gravitational potential energy (PE) gained by the water:
W = PE
The gravitational potential energy (PE) is given by the formula:
PE = m * g * h
Where:
m = mass of the water lifted
g = acceleration due to gravity (9.8 m/s^2)
h = height or altitude (2.00 km = 2000 m)
We are given the mass of the water (m) as 2.66 x 10^7 kg.
Substituting the known values into the formula, we get:
W = (2.66 x 10^7 kg) * (9.8 m/s^2) * (2000 m) = 5.23 x 10^11 J
Now, let's substitute the calculated work (W) and power (P) into the equation:
W = P * t
5.23 x 10^11 J = 2000 W * t
Solving for t, we get:
t = (5.23 x 10^11 J) / (2000 W)
t ≈ 2.615 x 10^8 seconds
Therefore, it will take approximately 2.615 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.
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 done by the pump and then divide it by the power of the pump. Here's how you can do it:
Step 1: Calculate the gravitational potential energy of the water vapor in the cloud.
The gravitational potential energy formula is given by PE = mgh, where m is the mass of the water vapor, g is the acceleration due to gravity, and h is the height.
Given:
Mass of water vapor (m) = 2.66 x 10^7 kg
Height (h) = 2.00 km = 2000 m
Acceleration due to gravity (g) = 9.8 m/s^2
PE = (2.66 x 10^7 kg) * (9.8 m/s^2) * (2000 m)
= 5.21 x 10^11 J
Step 2: Calculate the time required using the work-energy relationship.
The work done is equal to the change in potential energy. In this case, the work done by the pump will be equal to the gravitational potential energy of the water vapor.
Work done = PE = 5.21 x 10^11 J
Power of the pump = 2.00 kW = 2.00 x 10^3 W
To find the time (t), we use the equation: work done = power * time
time = work done / power
t = (5.21 x 10^11 J) / (2.00 x 10^3 W)
= 2.60 x 10^8 seconds
Therefore, it will take approximately 2.60 x 10^8 seconds for the 2.00 kW pump to raise the same amount of water to the cloud's altitude.