Help with a hydrodynamics problem

In summary, the conversation discusses the calculation of the distance from the bottom of a container, where a hole is punched at a certain height, to achieve the furthest horizontal distance for the water to shoot out. The conversation mentions the use of Bernoulli's Equation and the kinematic equation to solve for this distance, with the help of a member from a physics forum. The conversation also briefly touches on the use of super and subscripted numbers.
  • #1
MisterNi
4
0
Ok, I edited the post to show all my work, so here it is...

A hole is punched at a height h in the side of a container of height H. The container is full of water. If the water is to shoot as far as possible horizontally, (a) how far from the bottom of the container should the hold be punched? (b) Neglecting frictional losses, how far (initially) from the side of the container will the water land?

Here's what I have so far:

Using Bernoulli's Equation: P + .5*ρ*V^2 + ρ*g*h = a constant.
and since flow rate(Q) = A*V = constant, I set the equations at the opening of the tank = to the hole that's been punched which gives:

P1 + .5*ρ*V1^2 + ρ*g*h1 = P2 + V2^2 + ρ*g*h2

P1 and P2 are 0 since the pressures are atmospheric. .5*ρ*V1^2 is assumed as 0, since the cross sectional area of the tank is much greater than the cross sectional area of the punched hole. I took the initial point at the waterline on top of the tank so ρ*g*h1 is 0 which leaves me with:

.5*ρ*V2^2 = -ρ*g*h2 which simplifies to:

V2 = √[2*g*(H - h)] which is the velocity of the water just as it exists through the punched hole in the tank.

Using the kinematic equation:

d = V2*t + .5*a*t^2

t = √[(2*h)/g]

Here's where I'm stuck. I've tried V2 * t = d and using first order derivatives to determine the maximum...but I get exactly twice the answer in the back of the book which is (H/2). I've posted this problem on another board and a member suggested using the quadratic formula but I couldn't really follow the person's reasoning so I came to the Physics Forums for different suggestions

I'm sure (b) will be easy to find once I can figure out a way to solve for (a). Thanks.
 
Last edited:
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  • #2
MisterNi,
I think it's all OK until it comes to the kinematic equation.
Your equation is for the water shooting up, not sideways.
For shooting sideways, it is:
h = g/2 t2 and
d = v2t.
This yields
t = [squ](2h/g) and
t = d/v2
Eliminate t:
[squ](2h/g) = d/v2
Solve for d, substitute v2 with what you got, differentiate WRT to h, and find the maximum.
 
Last edited:
  • #3
Ah, thank you arcnets. The answer was staring me right in the face the entire time and I couldn't see it! Thanks again arcnets.

And off topic, how do you make super and subscripted numbers?
 
  • #4
You type

t [ sup ] 2 [ / sup ]

or

v [ sub ] 2 [ / sub ]

without the blanks to get t2, v2
 

What is hydrodynamics?

Hydrodynamics is the study of fluids in motion, including liquids and gases. This branch of physics deals with the principles of fluid mechanics, such as flow, pressure, and resistance.

What are some real-world applications of hydrodynamics?

Hydrodynamics has various applications in industries such as aerospace, marine engineering, and energy production. It is used to design and optimize ships, aircraft, and turbines, as well as to study weather patterns and ocean currents.

How do you calculate the force of a fluid on a surface?

To calculate the force of a fluid on a surface, you can use the formula F = ρAv², where ρ is the density of the fluid, A is the area of the surface, and v is the velocity of the fluid. This is known as the drag force and is commonly used in hydrodynamics problems.

What is Bernoulli's principle and how is it related to hydrodynamics?

Bernoulli's principle states that as the speed of a fluid increases, its pressure decreases. This principle is important in hydrodynamics as it explains the lift force on objects moving through a fluid, such as airplanes and boats.

What are some commonly used equations in hydrodynamics?

Some commonly used equations in hydrodynamics include the continuity equation, Euler's equation, and the Navier-Stokes equations. These equations help to describe the behavior of fluids in motion and are essential in solving hydrodynamics problems.

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