The Bernoulli equation is fundamental in fluid dynamics and describes the relationship between pressure, velocity, and height in an incompressible, non-viscous fluid. Here's an example to illustrate how it works.

### Problem:
Water flows through a pipe that changes in elevation and cross-sectional area. At point 1, the pipe has a diameter of 0.5 m, the velocity of water is 2 m/s, and the elevation is 10 m. At point 2, the pipe's diameter is 0.25 m, and its elevation is 5 m. Assuming the water density is 1000 kg/m³ and the flow is steady, find the velocity and pressure at point 2. The pressure at point 1 is 200 kPa.

#### Known values:
- Point 1:
  - Diameter: 0.5 m
  - Velocity: 2 m/s
  - Elevation: 10 m
  - Pressure: 200 kPa

- Point 2:
  - Diameter: 0.25 m
  - Elevation: 5 m
  - Velocity: ?
  - Pressure: ?

### Step 1: Continuity Equation (Conservation of Mass)
The continuity equation ensures that the mass flow rate is the same at both points, which implies:

\[
A_1 v_1 = A_2 v_2
\]

where \(A_1\) and \(A_2\) are the cross-sectional areas of the pipe at points 1 and 2, and \(v_1\) and \(v_2\) are the velocities.

The area of a circle is \(A = \pi \left( \frac{d}{2} \right)^2\).

- For point 1:  
  \[
  A_1 = \pi \left( \frac{0.5}{2} \right)^2 = 0.196 m^2
  \]

- For point 2:  
  \[
  A_2 = \pi

Explore an example of Bernoulli's equation in fluid dynamics, where we calculate the velocity and pressure at different points in a pipe. Given known values like diameter, velocity, and pressure, this problem uses Bernoulli's principle and conservation of mass to find the solution. Suitable for undergraduate engineering or physics students.

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