Homework Help: Classical mechanics - finding distance D in terms of velocity

1. Sep 14, 2011

Rumor

1. The problem statement, all variables and given/known data

"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."

2. Relevant equations

Equations of motion

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

F(tot) = ma

F(air) = -mbv

Weight = mg

3. 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.

Last edited: Sep 14, 2011
2. Sep 14, 2011

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?

3. Sep 14, 2011

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.

4. Sep 15, 2011

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.

5. Sep 16, 2011

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."