# Homework Help: Work-Energy Theorem with Line Integrals

1. Dec 6, 2012

### Vorde

1. The problem statement, all variables and given/known data

The problem is to prove the work-energy theorem: Work is change in kinetic energy.

2. Relevant equations

Line integral stuff, basic physics stuff.

3. The attempt at a solution

I'm given the normal definitions for acceleration, velocity and I'm given Newton's second law. I'm asked to show that $\int_c{F \cdot dr}$ leads to $\frac{1}{2}mv(b)^2-\frac{1}{2}mv(a)^2$ along an arbitrary path from a to b defined by $\vec{r}$(t).

I'm stuck. I got to $Work = m \int_c{ \frac{d \vec{v}}{d \vec{r}} \, \vec{v} \cdot \vec{v} \, dt}$, but I don't know how to proceed with that pesky dot product in the integrand, and without explicit functions to help me simplify, can anyone help?

Thanks a billion.

I'll post my work if it seems like I'm going the wrong way, but I don't know where I would be.

2. Dec 6, 2012

3. Dec 6, 2012

### Vorde

But without knowing an explicit formula for Force, how am I going to find a gradient field?

Also, I don't know that my Force field will be conservative, so I can't assume there will be a gradient curve.

4. Dec 6, 2012

### micromass

The statement of the result that you must prove makes me think that if a potential function $\phi$ exists, then $\phi(c(t))=\frac{mv(t)^2}{2}$. Maybe you can use this as definition of your potential function??

5. Dec 6, 2012

### micromass

It has to be conservative since the line integral will be the same regardless of the path you take.

6. Dec 6, 2012

### Dick

The force doesn't have to be conservative. It's still true. F=ma. a=dv/dt. v=dr/dt. So dr=v*dt. F*dr=m*(dv/dt)*(v*dt). Integrate it. You are making a great deal out of nothing.

7. Dec 6, 2012

### micromass

Listen to Dick and ignore my posts. I should have known better than to post in a thread that involves physics.

8. Dec 6, 2012

### Vorde

This is more or less what I wrote down, but can I get away with acting like those dot products are multiplications?

To Micro: Don't worry :)

9. Dec 7, 2012

### Dick

Not exactly. But if v is a vector then d(v.v)/dt=2*(dv/dt).v. You'll have to put in the dot products where I omitted them.

10. Dec 7, 2012

### Vorde

Ah, interesting, I can see where that would lead me. Thank you to the both of you :)