Why can projectile drag force be treated as depending only on velocity?

[shanname]

My classical mechanics textbook says that, for a projectile, the linear drag force is given by f = -bv and the second law is written as m$\ddot{r}$ = mg - bv (a second order differential equation) which can be rewritten as m$\dot{v}$ = mg - bv (a first order differential equation) because the forces depend only on v and not on r. But I can't figure out why this is the case. Doesn't v depend on r?

 

[mathman]

No. Velocity doesn't depend on location as such, only to the extent that location may effect some of the parameters.

 

[AlephZero]

$\mathbf v$ doesn't "depend" on $\mathbf r$ in the sense that it is some (unknown) function of $\mathbf r$ and probably some other variables as well.

The point is that $\mathbf v$ is just another name for $\mathbf{\dot r}$, (that's what "velocity" means!) and differentiating, $\mathbf{\dot v}$ is identically equal to $\mathbf{\ddot r}$.

 

[shanname]

Thank you, AlephZero. I believe I understand. I mean, I know that v is just another name for $\dot{r}$ and the like, I just thought it could be rewritten in terms of v for that reason. No one ever explained that this is true specfically because v did not "depend" on r... is it ever the case that v does depend on r?

 

[shanname]

sorry, didn't realize I wasn't bolding $\dot{r}$.