Question

Water from a vertical pipe emerges as a 16-cm-diameter cylinder and falls straight down 7.5 m into a bucket. The water exits the pipe with a speed of 2.2 m/s.

What is the diameter of the column of water as it hits the bucket?

Answer 1

Answer 2

To find the diameter of the column of water as it hits the bucket, we can apply the principle of conservation of energy.

When the water emerges from the pipe, it has a certain kinetic energy due to its speed. As it falls down and hits the bucket, its kinetic energy is converted into potential energy (gravitational potential energy) and is also dissipated as work done against air resistance.

The kinetic energy of the water as it exits the pipe is given by the formula:

KE = 0.5 * m * v^2

where m is the mass of the water and v is its speed. The mass of the water can be calculated using its density (ρ) and volume (V):

m = ρ * V

The volume of the water cylinder can be calculated using the formula for the volume of a cylinder:

V = π * r^2 * h

where r is the radius of the cylinder (half of the diameter) and h is the height of the cylinder.

Given that the diameter of the cylinder is 16 cm (which means the radius is 8 cm or 0.08 m) and the height (fall distance) is 7.5 m, we can calculate the volume V:

V = π * (0.08 m)^2 * 7.5 m ≈ 0.1508 m^3

Next, we need the density of water. The density of water at room temperature is approximately 1000 kg/m^3.

Now, we can calculate the mass of the water:

m = 1000 kg/m^3 * 0.1508 m^3 ≈ 150.8 kg

Now, we can calculate the kinetic energy (KE) using the given speed (v = 2.2 m/s):

KE = 0.5 * 150.8 kg * (2.2 m/s)^2 ≈ 181.42 J (joules)

When the water hits the bucket, its kinetic energy is fully converted into gravitational potential energy (GPE) at the height it falls (7.5 m) and work done against air resistance. The formula for gravitational potential energy is:

GPE = m * g * h

where g is the acceleration due to gravity (approximately 9.8 m/s^2).

Now, we can find the height h using the gravitational potential energy and the mass m:

GPE = 150.8 kg * 9.8 m/s^2 * 7.5 m ≈ 11,078.1 J

The total work done against air resistance is the difference between the initial kinetic energy and the final gravitational potential energy:

Work against air resistance = KE - GPE ≈ 181.42 J - 11,078.1 J ≈ -10,896.7 J

The negative sign indicates that work is done by air resistance to slow down the water.

When the water hits the bucket, it comes to rest, and its kinetic energy is fully converted into potential energy at the height it falls. At this point, its speed is zero, and the diameter of the column of water hitting the bucket is the same as the initial diameter, which is 16 cm.

So, the diameter of the column of water as it hits the bucket is approximately 16 cm.