Question

A liquid flowing from a vertical pipe has a very definite shape as it flows from the pipe. To get the equation for this shape, assume that the liquid is in free fall once it leaves the pipe. Just as it leaves the pipe, the liquid has speed v0 and the radius of the stream of liquid is r0.

Find an equation for the speed of the liquid as a function of the distance y it has fallen.

Combining this with the equation of continuity, find an expression for the radius of the stream as a function of y.

Answer 1

using bernoulli's principle,
\(P_1 + \rho g h_1 + \frac{1}{2} \rho v_1^2=P_2 + \rho g h_2 + \frac{1}{2} \rho v_2^2\)
where \(P_1=P_2=P_{atmosphere}\), and \(v_1=v_0\), and\(h_1-h_2=y\)
you'll get \(v_2^2=v_0^2+2gy\)
sub this into the \(A_1V_1=A_2V_2\) to get your function.