I'm currently taking a Diff Eq course, where the instructor has included physics problems into the homework. I love solving Diff Eq problems but physics problems make me grit my teeth and here's an example of something I was mildly stumped by today that will help show my frustration. 1. The problem statement, all variables and given/known data A ball whose mass is 1 kilogram rolls on a level surface, subject only to the force of friction. The initial velocity is 1 meter per second; after 1 second, the velocity is 0.8 meters per second. Assuming the magnitude of the frictional force is proportional to the velocity, (a) What is the friction constant? .... 2. Relevant equations I already know that frictional force is expressible as F_f = kv, k being the coefficient of static friction, and v the velocity. I also already know Newton's first and second laws (who doesn't?) sum(F_i) = F and F = ma. 3. The attempt at a solution Now, here is where my thinking started going crazy. I will example why in a moment, but first: I had noted that the initial force to send the ball into motion should be taken into account. Say F_0 = ma_0. Also, my diagram would show F_0 and F_f pointing in opposing directions, and that F_0 + F_f = F, by Newton's first law. Following that, I noted: At time t = 0, an instantaneous force F_0 = ma_0 is applied At time t > 0, a continuous force F_f = km is applied. Next, I began considering the piecewise function, to model the initial force: F_0(t) = m a_0 when t = 0 F_0(t) = 0 when t > 0 And I began carrying on in this very general, abstract away for the rest of the page. Giving an equation: F_0(t) + kv(t) = ma = mv'(t) But then I realized that this chapter is just supposed to be about separable equations, so something way simpler is required. However, I had a hard time convincing myself that something simpler could be used. This happened to be an odd problem, so rather than drive myself crazy with the way physics is worded, I just looked up the answer :( Here was my thinking process: The main problem I have with IVPs as a way to model certain things in physics is that it wants to model only instantaneous phenomenon, where dt is close enough to zero. But, as we all know, an impact does not occur in perfectly 0 seconds. Sometimes the brief moment of time is important to the actual result. What's worse, some "impacts" are actually forces that are applied for brief periods of time, and then released. Imagine a man pushing a box up a hill, stopping briefly, having the box slip a bit, and then resuming his push. How would that be modeled? So then, you, the veteran physicist, might say: Well, that's no longer one equation. It's more like a series of equations, marked off by discontinuous points when the forces are applied and released. It would be more like an algorithm then an equation. Well, I guess my bias is showing, as much of my love for mathematics is how much it resembles the logic of computer science. But having a general algorithm for solving force body diagrams would satisfy much of my insecurities on the subject. I'm not even asking to be given all of the numbers. In fact, I'm not even asking to be given the equations. I would at least like to know the relationship between the equations so I can begin filling in the details. More than anything, I want a rigorous proof that a set of equations and variables truly represent the solution to a problem. If I need to widdle down my model to something reasonable, at least let me know what those assumptions are, so I can write down in my problem "I took the assumption that ....".