First, a brief intro...I haven't done a differential equation since I was an undergrad (over 10 years ago!). I was tasked with finding the initial velocity of an object skidding to a stop over a surface including the frictional forces and air resistance.
Basically, I want to find a solution for V(x). Treating the vehicle as beginning at x=0 and moving in the postitive x direction a known distance D, the equation I came up with is
-fMg - kV2(x) = MV'(x)V(x)
The left side are the forces and the right side is mass times acceleration (acceleration was found by using the chain rule to put everything in terms of position x rather than time). f is the adjusted coeff. of friction, M is the object's mass, g is gravitational acceleration, k is a constant encompassing the other constants in front of v2 in the drag force.
I have attached my steps and final solution as a word document. Someone has already pointed out that I could have simplified it by combining constants more...so that I know. But, I'd like to see if my logic and my solution makes sense.
Thank you to anyone who can help.