How do I prove That Toricelli's equation, given here is correct?

[Aliskahir]

Homework Statement

The problem is shown here, Problem # 48

Homework Equations

I know that V1 = Sqrt of 2*G (h)

But the A1^2 / A2^2 part is unknown to me, and I've spent hours trying to find an explanation, but now I'm here...

Can anyone point me in the right direction, please?

 

[gneill]

Go back to Bernoulli's Principle. Find a relationship between V2 and V1 in terms of the areas. Substitute for V2 in Bernoulli.

 

[Aliskahir]

In Bernoulli's equation I found that

V1 = V2 * A2 / A1

 

[gneill]

In Bernoulli's equation I found that

V1 = V2 * A2 / A1

Solve that for V2 in terms of V1 (since you're looking to replace V2 in Bernoulli's equation.

Can you write the equation for Bernoulli's principle for this case?

 

[Aliskahir]

(Using D for Density)

If I replace V2 With the equation I get

1/2 DV1^2 + DGY1 = 1/2D (A1*V1/A2)^2 + DGY2

By the way, I appreciate your help, I'm lost mostly because I missed this particular day of class, and I'm trying to understand a friend's notes and the book. My test tomorrow is going to kill me =[

 

[gneill]

(Using D for Density)

If I replace V2 With the equation I get

P1 + 1/2 D V1^2 = P2 + 1/2 D (V1*A1 / A2) ^2

----hold on making revisions---

By the way, I appreciate your help, I'm lost mostly because I missed this particular day of class, and I'm trying to understand a friend's notes and the book. My test tomorrow is going to kill me =[

You're missing the potential energy term (D*g*h) on the left hand side. Remember, there's a height difference between the two locations. P1 and P2 are the ambient air pressure, which can be assumed equal at both locations, so they cancel. Solve for V1.

 

[Aliskahir]

1/2 DV1^2 + DGY1 = 1/2D (A1*V1/A2)^2 + DGY2

1/2 V1^2 (1-A1^2/A2^2) = G (Y2 - Y1)

V1 = Sqrt 2G(Y2-Y1) / (1-A1^2/A2^2)

 

[Aliskahir]

Yeah, I added it up top

 

[Aliskahir]

Is it... correct? or closer to being correct?

 

[gneill]

Is it... correct? or closer to being correct?

Can't you compare it with what is given in the problem statement?