Water from a vertical pipe emerges as a 10-cm-diameter cylinder and falls straight down 7.5 m into a bucket. The water exits the pipe with a speed of 2.0 m/s. What is the diameter of the column of water as it hits the bucket?

Verified step by step guidance

1. Step 1: Begin by analyzing the problem. The water exits the pipe with an initial speed of 2.0 m/s and falls vertically downward for a distance of 7.5 m. The goal is to determine the diameter of the water column when it reaches the bucket.
2. Step 2: Use the principle of conservation of mass, which states that the mass flow rate of water remains constant throughout its motion. The mass flow rate is given by \( \dot{m} = \rho A v \), where \( \rho \) is the density of water, \( A \) is the cross-sectional area, and \( v \) is the velocity.
3. Step 3: Calculate the velocity of the water as it hits the bucket using the kinematic equation \( v_f = \sqrt{v_i^2 + 2 g h} \), where \( v_i \) is the initial velocity (2.0 m/s), \( g \) is the acceleration due to gravity (9.8 m/s²), and \( h \) is the height (7.5 m).
4. Step 4: Relate the cross-sectional areas at the pipe and the bucket using the continuity equation \( A_1 v_1 = A_2 v_2 \). Here, \( A_1 \) is the area of the water column at the pipe, \( v_1 \) is the velocity at the pipe, \( A_2 \) is the area of the water column at the bucket, and \( v_2 \) is the velocity at the bucket.
5. Step 5: Solve for the diameter of the water column at the bucket using \( A = \pi (d/2)^2 \). Rearrange the continuity equation to find \( d_2 \), the diameter at the bucket, and substitute the values for \( v_1 \), \( v_2 \), and \( d_1 \) (10 cm).

Key Concepts

Continuity Equation

The Continuity Equation states that for an incompressible fluid flowing through a pipe, the product of the cross-sectional area and the fluid velocity remains constant. This principle implies that if the diameter of the pipe changes, the velocity of the fluid must adjust accordingly to maintain a constant flow rate.

Bernoulli's Principle

Bernoulli's Principle relates the pressure, velocity, and height of a fluid in steady flow. It indicates that an increase in the fluid's speed occurs simultaneously with a decrease in pressure or potential energy. This principle is crucial for understanding how the speed of water changes as it exits the pipe and falls into the bucket.

Kinematic Equations of Motion

Kinematic equations describe the motion of objects under constant acceleration. In this scenario, the water falling from the pipe experiences gravitational acceleration. These equations can be used to determine the final velocity of the water just before it hits the bucket, which is essential for calculating the diameter of the water column upon impact.