Understanding Bernouilli's Law: Calculating Distance and Depth in Water Tanks

[mousesgr]

A tank is filled with water to a height H. A hole is punched in one of the walls at a depth h below the water surface. (a) Show that the distance x from the base of the tank to the point at which the resulting stream strikes the floor (b) Could a hole be punched at another depth to produce a second stream that would have the same range? If so, at what depth? (c) At what depth should the hole be placed to make the emerging stream strike the ground at the maximum distance from the base of the tank?

i know the ans of (a) is x = 2[h(H-h)] ^1/2, i want to ask...how to prove
(b) & (c) also

 

[e(ho0n3]

Don't you need the velocity of the water escaping through the hole before you can solve any of the problems?

 

[marlon]

what about Bernouilli's law or the continuity law...

marlon

 

[e(ho0n3]

Still, don't you need the velocity at which the height of the water in the tank is decreasing? Unless the tank is big enough so that you can effectively set this to zero, I don't see how to go about solving part (a) by just knowing heights.

 

[Doc Al]

Unless the tank is big enough so that you can effectively set this to zero...

Bingo! That's the typical assumption.

 

[e(ho0n3]

Bingo! That's the typical assumption.

OK. So given this assumption we have:

$$\rho g H = \rho v^2 + \rho g (H-h)$$

right? Then I get for x:

$$x = \sqrt{2(H-h)(H+h)}$$

This is different from what mousesgr got. Is this right?

 

[Doc Al]

OK. So given this assumption we have:

$$\rho g H = \rho v^2 + \rho g (H-h)$$

right?

You forgot a factor of 1/2. The efflux speed is $v = \sqrt{2 g h}$.

 

[e(ho0n3]

Oh yeah. You'll have to forgive me. I'm a little rusty (actually I'm rusted out but anyways). So the answer to part (b) would be no and for (c) we would just maximize x using calculus.

 

[arildno]

Still, don't you need the velocity at which the height of the water in the tank is decreasing? Unless the tank is big enough so that you can effectively set this to zero.

Note also that if you couldn't do this approximation, you couldn't use the approximation of STATIONARY FLOW.
(the domain would change in time, and hence, the velocity would change locally in time as well..)

 

[mousesgr]

You forgot a factor of 1/2. The efflux speed is $v = \sqrt{2 g h}$.

i know how to find the velocity, but how to find x?

 

[mousesgr]

i know how to find the velocity, but how to find x?

i know how to do part(A)
thx everybody

 

[mousesgr]

Oh yeah. You'll have to forgive me. I'm a little rusty (actually I'm rusted out but anyways). So the answer to part (b) would be no and for (c) we would just maximize x using calculus.

part (b) is no?
why?

 

[e(ho0n3]

Look at your expression for x. If (b) is yes, then there are two values of h, say h1 and h2 such that x(h1) = x(h2) and h1 != h2. Just from looking at the expression I can tell there are no such h1 and h2 (but I'm no math god so maybe I'm totally wrong). If you're unsure, graph x and find out.

 

[Doc Al]

For part b: Use the answer for part a and solve it backwards. Looks like a quadratic equation to me.

 

[mousesgr]

For part b: Use the answer for part a and solve it backwards. Looks like a quadratic equation to me.

by observation...the ans is h

 

[Doc Al]

by observation...the ans is h

What does that mean? A little voice spoke to you?

Just teasing... Why don't you try actually solving it now. Let $h_1$ be your original height h, so $x = 2\sqrt{h_1(H-h_1)}$. Now solve the quadratic equation $x^2 = 4h(H-h)$ for h. The quadratic will have two solutions. (One solution is $h = h_1$, of course. But what's the other?)