You raise a bucket of water from the bottom of a deep well.
If your power output is 128 W, and the mass of the bucket and the water in it is 6.00 kg, with what speed can you raise the bucket? Ignore the weight of the rope.
The speed with which you can raise the bucket is equal to the power output divided by the mass of the bucket and water, or 128 W / 6.00 kg = 21.33 m/s.
To solve this problem, we can use the work-energy principle. The work done on the bucket is equal to the change in its kinetic energy.
1. Determine the work done by the power output:
Power = Work / Time
Work = Power * Time
2. Calculate the time taken to raise the bucket:
Work = Force * Distance
Distance = Velocity * Time
Work = Force * (Velocity * Time)
Given that the weight of the bucket is equal to its mass times the acceleration due to gravity (9.8 m/s^2):
Force = Weight
Force = Mass * Acceleration due to gravity
Work = (Mass * Acceleration due to gravity) * (Velocity * Time)
Plugging in the weight of the bucket and the value of work from step 1:
(Mass * Acceleration due to gravity) * (Velocity * Time) = Power * Time
3. Solve for the velocity:
Mass * Acceleration due to gravity * Velocity = Power
Rearranging the equation:
Velocity = Power / (Mass * Acceleration due to gravity)
Plugging in the given values:
Velocity = 128 W / (6.00 kg * 9.8 m/s^2)
Now we can calculate the velocity:
Velocity = 128 W / (6.00 kg * 9.8 m/s^2)
Velocity = 128 / (6.00 * 9.8) m/s
Calculating the value:
Velocity ≈ 2.15287 m/s
Therefore, the speed at which you can raise the bucket is approximately 2.15 m/s.
To determine the speed at which you can raise the bucket, you need to apply the work-energy principle. According to this principle, the work done by a force is equal to the change in kinetic energy. In this case, the work done by your power output is equal to the change in kinetic energy of the bucket and the water.
The work done by a force is given by the equation:
Work = Force × Distance × cos(θ)
In this scenario, the force is equal to the weight of the bucket and the water since they are being lifted vertically. The distance is the height of the well and the angle θ is 0 degrees because the force and displacement are in the same direction.
Therefore, the work done is given by:
Work = Weight × Height × cos(0°)
Since the work done is equal to the change in kinetic energy, we can write:
Work = ΔKE
Now, the kinetic energy of an object is given by:
KE = (1/2) × mass × velocity^2
By substituting the formula for kinetic energy into the equation for work, we get:
(1/2) × mass × velocity^2 = Weight × Height × cos(0°)
Rearranging the equation, we can solve for the velocity:
velocity = sqrt((2 × Weight × Height × cos(0°)) / mass)
However, the weight of an object is given by:
Weight = mass × gravity
Substituting this into the equation, we get:
velocity = sqrt((2 × mass × gravity × Height × cos(0°)) / mass)
Since the mass of the bucket and water is given as 6.00 kg, and gravity is approximately 9.8 m/s^2, we can simplify the equation to:
velocity = sqrt((2 × 6.00 kg × 9.8 m/s^2 × Height) / 6.00 kg)
Simplifying further, we have:
velocity = sqrt(2 × 9.8 m/s^2 × Height)
Now, given that your power output is 128 W, we can use the formula for power:
power = Work / time
Since you want to find the speed, we can rearrange the formula to solve for time:
time = Work / power
Substituting the formula for work and known values, we have:
time = [(1/2) × mass × velocity^2] / power
Plugging in the known values, our equation becomes:
time = [(1/2) × 6.00 kg × velocity^2] / 128 W
Finally, solving for velocity:
velocity = sqrt((2 × power × time) / mass)
To calculate the exact speed with which you can raise the bucket, you need to know the time it takes to raise the bucket.