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.

 

[Orodruin]

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]

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

No it is not. $\vec v^2 = \vec v \cdot \vec v = v^2$.

 

[archaic]

View attachment 240807
View attachment 240808
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.

Could you tell me the book's name?

 

[JD_PM]

Could you tell me the book's name?

Sure

Vector Analysis; Schaum's outlines

 

[ehild]

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:
$F=m\frac{d v}{d r} \frac{d r}{d t}=m v \frac{d v}{d r}=\frac{m}{2}\frac{d(v^2)}{dr}$ .