First order non-linear differential equation

[cen4]

Homework Statement

Hello, I was given an extension problem in a Dynamics lecture today and am struggling to solve it.

It is a simple scenario: a particle of mass m is accelerating due to Galilean gravity, but is subject to a resistive force that is non-linear in the velocity of the particle. This is in the usual Cartesian coordinate system, where z is the unit vector pointing vertically.

I will use bold font to denote vectors, and let v' = dv/dt [not the usual notation, but I do not know how else to easily show derivatives on forums].
Also, v is the speed of the particle, magnitude of velocity v.

Homework Equations

The resistive force is given by : -μvv

Where μ is a constant.

From N II : mv' = -gz - μvv

The Attempt at a Solution

I'm unsure on how to solve this non-linear ODE. My attempt at a solution via separation of variables the equation ended up with a solution involving arctan. However, I was confused about the idea of integrating with respect to v, the vector, particularly when we have z involved - so this solution may be completely invalid. We were told not to split up the differential equation into components of the vectors, but instead to solve completely through as the problem was given.

Any help would be appreciated, thanks.

 

[CAF123]

In the case of quadratic resistance and an arbritary motion of the body in the plane (e.g not constrained to move horizontally or vertically), then the equations of motion for the velocity in the x and y direction do not decouple.

If you separated variables, did you consider the motion to occur in one direction?