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When the lid is not kept on and a hole is created about an inch or two from the bottom the water bottle, I observed that the water flows out the bottle in a parabolic manner. What is the reasoning behind this mechanism?

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Consider it as a stream of particles coming out from the hole Now you can relate it with a particle having some initial velocity (water inside pushes water at the corner with a force) You can apply the equation of motion to find the time period and height.


$H=\frac{1}{2} gt^2$, h-height of hole from ground

Range=$ut$


So you see it will have a parabolic path similar to a projected particle when the particle is at its maximum height

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