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I'm looking to calculate the initial velocity of a large object sliding to a stop over a surface. This is easy without the drag force of the air, but I'd like to include that to prove a point in my research. As a quick background, I haven't done a differential equation in over a decade since I was an undergrad, so I'm trying to relearn on the fly.

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 - kV^{2}(x) = MV'(x)V(x)

The left side are the forces and the right side is mass times acceleration. 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 v^{2}in the drag force.

I want to find the velocity V with respect to position x. The solution I came up with is:

[itex]V(x) = \sqrt{-\frac{Mfg}{k}(1-e^{\frac{-2k(x-D)}{M}})}[/itex]

This uses the fact that when x = D, V = 0. Does this look correct? My calculations make sense, but I want to be sure. I can include the intermediate steps if necessary, but that's a lot of typing!

Thanks from an old out of practice former physics nerd who now only uses this stuff on occasion.

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# Initial speed of object sliding to a stop w/ air resistance

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