Question

A large storage tank with an open top is filled to a height h0. The tank is punctured at a height h above the bottom of the tank. Find an expression for how far from the tank the exiting stream lands. (Let d be the horizontal distance the stream of water travels. Use any variable or symbol stated above as necessary. ? for the density of water and g. Do not substitute numerical values; use variables only.)

Answer 1

Bernoulli equation for invicid flow:
P1/rho + v1^2/2 + g*h1 = P2/rho + v2^2/2 + g*h2

Location 1 is at the top of the tank, location 2 is at the punctured hole.
Assume atmospheric pressure at top of tank, and obviously, once outside the hole, the pressure is atmospheric.

Therefore, pressure effects are not part of the solution.
v1 = 0, because we neglect the top of the tank velocity

Our equation reduces to:
g*h1 = v2^2/2 + g*h2

Solve for v2:
v2 = sqrt(2*g*(h1 - h2) )

This should look familiar, even in the field of ordinary mechanics.

To apply ballistic motion, the stream travels horizontally initially. Use the vertical descent to find the time required. Initially, velocity in the vertical direction is zero. Therefore, the following applies:

h1 = 1/2*g*t^2

solve for t:
t = sqrt(2*h1/g)

Constant velocity motion in the horizontal direction applies. Therefore:
d = v2*t

Substitute:
d = sqrt(2*g*(h1 - h2)*2*h1/g)

Simplify:
d = 2*sqrt((h1-h2)*h2)