I love science and am decent with math, however my lack of skills with advanced calculus is a little embarrassing. I am in awe at some of the skills shown by some of the people here. So here is my query.

There was a freefall acceleration problem discussed elsewhere in this forum and I jumped in to correct a few errors made by some who answered. While I was proving to myself that air resistance was indeed a major factor in said problem I found that I was stumped on another aspect of the motion.

While it was quite simple to verify that the terminal velocity of the object would be less than one half that suggested by freefall equations by including air resistance formulas and finding when they were equal to the force of gravity, I could not find the solution to where and how fast the object was moving with respect to time taking air resistance into account at any time other than at terminal velocity.

When the acceleration is constant as it is with gravity for small changes in altitude, single or double integration can be done to find velocity or position function with no problem by me. My sticking point occurs when the force is varying.

Here is my example:

∑F = ma = pi(density of material)r^2(v^2) - mg

a = {[3 (density of material) v^2] / [4r(density of medium)]} - g

This equation is using the assumption that the object is perfectly smooth and round. So you can see that a varies with time as well as velocity and position. The only variable in the a function is that of velocity.

So I finally shut up, can someone please explain how to take a double integration of a varying force? Thank you for your help.

tap,tap,tap...."Is this thing on?"

Just kidding. Perhaps someone here could direct me to a good resource for working on differential equations. Maybe I was asking for something a little too demanding earlier.