# Particle moving in conservative force field

#### JD_PM

I don't get why $F \cdot dr = \frac{mv^2}{2}$

I know this has to be really easy but don't see it.

Thanks.

Related Calculus and Beyond Homework News on Phys.org

#### Orodruin

Staff Emeritus
Homework Helper
Gold Member
2018 Award
What is the derivative of $\vec v^2$ with respect to time?

#### JD_PM

I see what you mean but that is the vector and not the magnitude

#### Orodruin

Staff Emeritus
Homework Helper
Gold Member
2018 Award
I see what you mean but that is the vector and not the magnitude
No it is not. $\vec v^2 = \vec v \cdot \vec v = v^2$.

#### JD_PM

Could you tell me the book's name?
Sure

Vector Analysis; Schaum's outlines

#### ehild

Homework Helper
I think the author intended to use
$F= \frac{m}{2}\frac{d(v^2)}{dr}$.
It comes from the definition of force
$F=m\frac{d^2r}{dt^2}=m\frac{dv}{dt}$
as $v=\frac{dr}{dt}$.
Applying chain rule when using derivative with respect to r instead of t:
$F=m\frac{d v}{d r} \frac{d r}{d t}=m v \frac{d v}{d r}=\frac{m}{2}\frac{d(v^2)}{dr}$ .

"Particle moving in conservative force field"

### Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving