A cylinder of height H and open to the atmosphere is full of water and standing upright on a table. There are three holes in the side of the cylinder, although these are covered to start with. One hole is 1/4 of the way down from the top, while the other two are 1/2 and 3/4 of the way down.

Which hole shoots the water furthest horizontally? Start with a hole a distance h from the top of the cylinder. With what speed does the water emerge from the hole?

Let's apply Bernoulli's equation:

ρgy₁ + ½ρv₁² + P₁ = ρgy₂ + ½ρv₂² +P₂

Point 2 can be just outside the hole, so the pressure is atmospheric pressure. We're looking for v₂. Point 1 can actually be any point inside the cylinder, although some points are more convenient to work with than others. Good choices would be a point at the same level as the hole, where the pressure is atmospheric pressure + ρgh, or a point at the top of the cylinder where the pressure is atmospheric pressure. Let's try a point at the top of the cylinder.

Measuring y's from the level of the hole gives:

ρgh + ½ρv₁² + P_atm = ½ρv₂² + P_atm

Cancelling the pressures, and then the factors of density, we're almost done:

gh + ½v₁² = ½v₂²

This is a good time to bring in the continuity equation:

A₁v₁ = A₂v₂

The area of the hole is much less than the area of the cylinder, so we will simply assume that v₁ is negligible compared to v₂. This gives:

gh = ½v₂²

so v₂ = [2gh]^½

This should look familiar to you.

The rest of the analysis involves recognizing that the water emerges from the hole with an initial horizontal velocity, and applying projectile motion equations to determine the horizontal distance reached by the water stream.

We can use the y-information (vertical) to get time. The initial velocity in the y-direction is zero, and the distance traveled is H-h. This gives:

H-h = ½gt²

The time it takes to reach the ground is:

t

=

[

2 (H-h) g

]

½

The distance traveled in the x-direction (horizontal) in this time is:

x = vt = (2gh)½

[

2 (H-h) g

]

½

= 2 [h(H-h)]½

The equation tells us that the horizontal distance traveled by water coming from a hole a distance h down the cylinder is the same as that for water coming from a distance h from the bottom (H-h from the top).

It is also easy to show that x is maximized when h = H/2. In other words, water coming from a hole halfway down the cylinder goes furthest horizontally.