Final answer:
To calculate the gauge pressure required for water to emerge from the small end of a pipe with a speed of 12 m/s when elevated 8 m above the large end, we can use Bernoulli's equation. By considering the potential energy term in Bernoulli's equation and equating it to the difference in pressure due to elevation, we can solve for the gauge pressure.
Step-by-step explanation:
To calculate the gauge pressure required for water to emerge from the small end of the pipe with a speed of 12 m/s when the small end is elevated 8 m above the large end, we can use Bernoulli's equation.
Bernoulli's equation states that the sum of the pressure, kinetic energy, and potential energy per unit volume is constant along a streamline.
Since the small end is elevated, we need to consider the potential energy term in Bernoulli's equation. Assuming the pressure at the large end is atmospheric pressure, we can calculate the gauge pressure at the small end by equating the potential energy term to the difference in pressure due to elevation:
P + 1/2 ρv^2 + ρgh = P0
where P is the pressure at the small end, ρ is the density of water, v is the speed of water at the small end, g is the acceleration due to gravity, h is the elevation difference, and P0 is atmospheric pressure.
Substituting the given values, we can solve for the gauge pressure:
P = (1/2 ρv^2 + ρgh) - P0