Here is the exact problem:
Lawn Sprinkler: A lawn sprinkler is constructed in such a way that is constant, where y ranges between 45(degrees) and 135(degrees). The distance the water travels horizontally is:
x = (V^2 sin(2y))/32
where v is the speed of the water. Find and explain why the water does not water evenly. What part of the lawn receives the most water?
Here is the image.
If anyone could walk me through this, that would be awesome!
I've got a question about this problem.
Every solution I see says the most water will fall where , which means . Since is the horizontal distance the water travels, this is the angle at which the horizontal distance the water travels is maximized/minimized. But shouldn't most water fall where the velocity is minimized? Shouldn't we be looking at where is minimized? In other words, shouldn't we be looking at where ?
If that is the case, then I get and . This seems correct because in my experience these types of sprinklers water most at the sprinkler and least out farthest from the sprinkler.
But then every other person's solution that I have seen is wrong. I'm not trusting that! Am I missing something?
x represents the position, or point where the water falls on the ground relative to the sprinkler, not a distance
dx/dt is the rate of change of that position, i.e. velocity of the point where the water hits the ground.
when $\bigg|\dfrac{dx}{dt}\bigg| \to 0$ at the endpoints of the oscillation, the amount of water hitting the ground is increased because the water flow being constant has more time to hit the ground ... as $\bigg|\dfrac{dx}{dt}\bigg|$ reaches a maximum at $\theta=\dfrac{\pi}{2}$, the position where the water hits the ground moves fastest, less time for the constant water flow to hit the position close to the sprinkler.
dx/dt is a minimum at $\theta = \dfrac{\pi}{4}$ and $\theta = \dfrac{3\pi}{4}$ ... note the attached graph. Red indicates velocity to the left from , blue to the right.
Of course, all of this is based on the position model presented in this problem. There may well be oscillating sprinklers that are designed differently. Check out the link for a very detailed analysis on oscillating sprinklers.
Design of an Oscillating Sprinkler.pdf