Where the effect of air resistance is small, the path of projectile is followed when a fluid discharged through an orifice into the air
The initial velocity of the jet is

V₀ = C_v?2gh

where h= head on center line of orifice, ft (m),
C_v =coefficient of velocity.

The direction of the initial velocity depends on the orientation of the surface in which the orifice is located.The velocity of the jet in the X direction (horizontal) remains constant

V_x = V₀ = Cv ?2gh

The velocity in the Y direction is initially zero and there after a function of time and the acceleration of gravity:

V_y = gt

The X coordinate at time t is
X = V_x t = tC_v ?2gh

The Y coordinate is

Y = V_avg t= gt²/2

where V_avg =average velocity over period of time t.

The equation for the path of the jet:

X₂ = C_v* C_v *4hY

When the inflow is less than the outflow then that condition is called the falling head condition.The time required for a certain quantity of water to flow from a reservoir can be calculated by equating the volume of water that flows through the orifice or pipe in time dt to the volume decrease in the reservoir. If the area of the reservoir is constant,

t=2A (?h₁ -?h₂)/ (Ca ?2g)

where h₁= head at the start, ft (m)

h₂ =head at the end, ft (m)

t =time interval for head to fall from h₁ to h₂,(s )

Orifice Discharge into Free Air

An orifice is an opening with a closed perimeter through which water flows. Orifices may have any shape, although they are usually round, square, or rectangular.
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Energy losses occur in pipe contractions, bends, enlargements, and valves and other pipe fittings. These losses can usually be neglected if the length of the pipeline is greater than 1500 times the pipe diameter. However, in short pipe- lines, because these losses may exceed the friction losses, minor losses must be considered.
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Darcy Weisbach formula
Darcy Weisbach formula which is valid for laminar or turbulent flow in all fluids is one of the most commonly used formula for determining the head loss.

h_f=[fLV²]/2gD

where
h_f=head loss due to friction, ft (m)

f= friction factor

L =length of pipe, ft (m)

D= diameter of pipe, ft (m)

V =velocity of fluid, ft/s (m/s)

g =acceleration due to gravity, 32.2 ft/s2 (9.81 m/s2)

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