# Proving a known function of position via Chain Rule

#### kylera

1. Homework Statement
Use the Chain Rule to prove that for rectilinear motion, when the acceleration is a known function of position, you can find the velocity as a function of position via the integral

$$\frac{v^{2}-v_{0}^{2}}{2} = \int^{s}_{s_{0}}a(s)ds$$

2. Homework Equations
$$v^{2}=v_{0}^{2}\times2as$$

3. The Attempt at a Solution
I took the left fraction, substituted v^2, simplified and got $$as$$. I let A be as, then took dA to get a da. Now I'm stuck.

Related Introductory Physics Homework Help News on Phys.org

#### ace123

I think you simplified wrong. It's a multipication sign not addition. See what I mean?

#### kylera

Hold up, I wrote the question on the board wrong -- it is supposed to be a plus for the relevant equation part.

#### calef

I'm not exactly sure what they're asking here. For constant accelerations, you "relevant equation" is basically the answer, assuming you swap out the "x" for a "+" and take a square root. What's throwing me is the request for proof by chain rule.

#### tiny-tim

Homework Helper
Chain Rule

1. Homework Statement
Use the Chain Rule to prove that for rectilinear motion, when the acceleration is a known function of position, you can find the velocity as a function of position via the integral

$$\frac{v^{2}-v_{0}^{2}}{2} = \int^{s}_{s_{0}}a(s)ds$$

2. Homework Equations
$$v^{2}=v_{0}^{2}\times2as$$

3. The Attempt at a Solution
I took the left fraction, substituted v^2, simplified and got $$as$$. I let A be as, then took dA to get a da. Now I'm stuck.
Hi kylera! You were asked to use the Chain Rule. So …

Hint: the LHS is ∫vdv. So use the Chain Rule on dv. 