Okay, first a brief intro. I studied math/physics extensively as an undergrad...but that was over 10 years ago now. My day job keeps me doing more basic physics and math regularly, but I haven't, for example, solved a differential equation in over a decade! Anyway...(adsbygoogle = window.adsbygoogle || []).push({});

I'm trying to calculate the initial velocity of an object sliding to a stop over a single level surface with a known friction coefficient and a known distance (D). This is easy ignoring air resistance, but I want to factor that in. I'm setting up my forces so that the object is traveling in the positive x direction (therefore, friction and drag are negative).

The F = ma equation that I came up with is

F = -fMg - kv^{2}(x) = ma = mV'(x)V(x)

where V(x) is the velocity as a function of x, f is the coeff. of friction, M is the object's mass, g is gravitational acceleration, k is a constant representing all the other constants in the drag equation before the v^{2}(cross sectional area, density, drag coeff).

Solving for v(x) while breaking out my old diffy q book and looking for help on the internet (and having substitution/chain rule flashbacks), I came up with the following:

V(x)=Sqrt[ (-Mfg+Mfge^{(-2kx+2kd)/M}) / k]

I know I skipped a lot of steps, but it's a lot of typing! I can include more if needed, but I'm curious if I'm on the right track. I used the fact that when x=d, v=0 (end of deceleration) to solve it. The numbers vs. ignoring drag make sense when I do the calculations.

Any help/advice would be appreciated.

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# Speed of object sliding to a stop w/ air resistance with respect to position

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