Confused about equation solving

[JesseK]

Hey!

I'm confused about this:

½v²₁ = ½v²₀ - gh₁

v²₁ = v²₀ - 2gh₁

v₁ = √v₀² - 2gh₁

Soo, what basically happened here when solving the equation for v1? I understand that the equation should be multiplied by ½ and then take a square root in order to solve for v1, but what's that factor "2" before gh (bold text). Where did that come from?

Thank you!

 

[Ibix]

I understand that the equation should be multiplied by ½

No, you are multiplying by 2. Does that answer your question?

 

[berkeman]

As @Ibix says, multiply both sides of the equation by 2 (so the equal sign still holds -- always do the same operation on both sides of the equation when simplifying it):

$$2(\frac{1}{2}v_1^2) = 2(\frac{1}{2}v_0^2 - gh_1)$$

 

[Ibix]

1/2 * 1/2 = 1

What? If I have an apple and I cut it in half, and cut one if the halves in half, you expect the half of a half to be a whole apple?

 

[JesseK]

What? If I have an apple and I cut it in half, and cut one if the halves in half, you expect the half of a half to be a whole apple?

Yeah, it was my bad, sorry. Obviously 1/2 * 1/2 = 1/4 haha :D

 

[JesseK]

As @Ibix says, multiply both sides of the equation by 2 (so the equal sign still holds -- always do the same operation on both sides of the equation when simplifying it):

$$2(\frac{1}{2}v_1^2) = 2(\frac{1}{2}v_0^2 - gh_1)$$

Oh, so every term on both sides of the equation should be multiplied by 2 (those brackets)? Now, I gotcha, thank youu! :D

 

[Mark44]

Hey!

I'm confused about this:

½v²₁ = ½v²₀ - gh₁

v²₁ = v²₀ - 2gh₁

v₁ = √v₀² - 2gh₁

The line above confused me, at first, since it looks like you're taking the square root only of the $v_0^2$ term. You really should be using parentheses, like this: v₁ = √(v₀² - 2gh₁)

Also, mathematically speaking, there will be two solutions for $v_1$:
v₁ = ±√(v₀² - 2gh₁)

Soo, what basically happened here when solving the equation for v1? I understand that the equation should be multiplied by ½ and then take a square root in order to solve for v1, but what's that factor "2" before gh (bold text). Where did that come from?

Thank you!