Classical mechanics - finding distance D in terms of velocity

[Rumor]

Homework Statement

"A passenger (mass m) initially at rest steps out of an airplane. Assume down is the positive x-axis and put the origin at the airplane. Assume the air resistance force is linear in the velocity so F(air)= -mbv. Find the distance D he has fallen when his velocity is v."

Homework Equations

Equations of motion

The "vdv/dx trick": d²z/dt² = (z-dot)*(d(z-dot)/dz)

F(tot) = ma

F(air) = -mbv

Weight = mg

The Attempt at a Solution

Here's how far I've gotten:

Since the skydiver is only falling in the x direction, there's only one equation of motion, which I found to be ma = -mbv + mg [or, alternatively, m(x-double dot) = -mb(x-dot) + mg]. Now, I know I want the relation of distance and velocity, without time, so I use the "vdv/dx" trick (so that there's no longer time in the equation).

That makes this mv*(dv/dx) = -mbv + mg, or m(x-dot)*(d(x-dot)/dx) = -mb(x-dot) + mg. I rearranged this to get dx = (-m/b)*(vdx/v-(mg/b)), where -(mg/b) is the terminal velocity.
I'm sorry for all the writing, but am I correct so far? And how do I continue to solve the problem from here? Any help would be appreciated.

 

[kuruman]

First off, divide through by m to simplify the expression a bit. Secondly, can you separate variables, i.e have things with "v" on one side and "x" on the other?

 

[daveb]

First, since m is common to every term, I would eliminate that to make things easier. Next, you have vdv/dx = g-bv. Try to isolate v and dv on one side, and dx on the other side.

 

[Rumor]

Alright, so...

I went back and simplified what I had first and ended up with v(dv/dx) = -bv - g. So, after isolating dz, I end up with dz = -(vdv/(bv+g)).

Now, I know my next step is to integrate this, but I'm not sure what the limits would be.

 

[kuruman]

The problem tells you what the limits should be.

"A passenger (mass m) initially at rest steps out of an airplane. Assume down is the positive x-axis and put the origin at the airplane. Assume the air resistance force is linear in the velocity so F(air)= -mbv. Find the distance D he has fallen when his velocity is v."