Bernoulli's equation book problem

[bobsmith76]

Homework Statement

This book is skipping over too many steps

where A₁ = 1m²
A₂ = .5 m²

y₁ - y₂ = 5m

The Attempt at a Solution

I'm having trouble getting from one step to another.

By my reckoning this equation

can be simplified by removing P₀ and ρ since they are the same on both sides of the equation. Then we can multiply each term by 2.

That makes

v1² + 2gy₁ = ((A₁/A₂)v₂)² + 2gy₂

Now I subtract 2gy₁ from both sides. That makes:

v1² = ((A₁/A₂)v₂)² + 2gy₂ - 2gy₁

The book is showing that that is a mistake. I can't figure out why.

Also if I subtract ((A₁/A₂)v₂)² from both sides that would make (ignoring 2gy)

v1² - ((A₁/A₂)v₂)²

But the book has it as

v1² multiplied by (1- ((A₁/A₂)v₂)²

Why?

 

[Doc Al]

can be simplified by removing P₀ and ρ since they are the same on both sides of the equation. Then we can multiply each term by 2.

OK.

That makes

v1² + 2gy₁ = ((A₁/A₂)v₂)² + 2gy₂

Shouldn't that be v₁ on the right hand side, not v₂?

 

[bobsmith76]

Yea, I guess it should but I'm still stuck with my two questions.

 

[Doc Al]

Yea, I guess it should but I'm still stuck with my two questions.

First change that v₂ to a v₁ and your questions may go away.

 

[bobsmith76]

Ok, I understand the part about v1 and v2, but would about

2gy₂ - 2gy₁

 

[emailanmol]

y2-y1 is the height difference between the two points under consideration.

 

[Doc Al]

Ok, I understand the part about v1 and v2, but would about

2gy₂ - 2gy₁

What about it? Realize that your book is calling Δy = h.

 

[bobsmith76]

In the final step it should be

√h

not

√(2gh)

Because the 2g - 2g = 0

 

[Doc Al]

In the final step it should be

√h

not

√(2gh)

Because the 2g - 2g = 0

The '2g' factor doesn't cancel!

Example: 5a - 3a = 2a, not 2.