Final answer:
To accelerate water down a vertical pipe at 5 m/s^2, the pressure gradient along a streamline is 49000 Pa/m based on Bernoulli's Equation.
Step-by-step explanation:
To determine the pressure gradient along a streamline that will accelerate water down a vertical pipe at 5 m/s^2, we can use Bernoulli's Equation. In this case, the acceleration is due to gravity, so we need to compare the pressure at two different points along the streamline.
The pressure gradient is given by dp/ds = ρgh, where ρ is the density of water, g is the acceleration due to gravity, and h is the vertical distance between the two points.
Given that water is accelerating at 5 m/s^2, we can substitute this value for h in the equation. The density of water is approximately 1000 kg/m^3 and the acceleration due to gravity is approximately 9.8 m/s^2.
Substituting these values into the equation, we get dp/ds = (1000 kg/m^3)(9.8 m/s^2)(5 m/s^2) = 49000 Pa/m.
Therefore, the correct answer is a) 49000 Pa/m.