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First, let me preface this question that I have been out of academia for about 5 years now, and am just starting to get back into it, although it seems my Calc/Kinematics is a little rusty.

b]1. Homework Statement [/b]

An object of weight W is falling through a medium such that the object's drag force is proportional to its velocity. Express the velocity in terms of time if the initial velocity of the object is zero.

F=ma

F=KV

a=dV/dt

Ok, so just from the problem statement alone, I know we have an object on which two forces are acting; the weight of the object due to gravity and the drag force on the object.

Fnet = mg - KV

Since g is always a constant, it's the acceleration of the drag force we are trying to solve for, correct? Since F=KV, and F=ma, by association we have ma=KV, or a=KV/m, so now we have a function of acceleration in terms of velocity. If we plug this into a=dV/dt, we end up with:

KV/m = dV/dt

or

dt = (m/KV)dV

Integrating both sides, we end up with:

[itex]\int[/itex]dt = (m/K)[itex]\int[/itex](1/V)dV or t = (m/K)ln(V)+C

Now if we solve for V in terms of t we get:

ln(V) = (tK/m)-C or V = e^[(tk/m)-C]

So this is as far as I have been able to get with this problem, because when you plug in initial velocity to solve for C, you get 0 = e^(0-C), but e^x is undefined when x = 0.

Am I making a wrong assumption at the beginning of the problem? I've got about 6 pages of scratch work and I feel this is the closest I've gotten to the actual solution. Any advice would be helpful. Thanks in advance.

b]1. Homework Statement [/b]

An object of weight W is falling through a medium such that the object's drag force is proportional to its velocity. Express the velocity in terms of time if the initial velocity of the object is zero.

## Homework Equations

F=ma

F=KV

a=dV/dt

## The Attempt at a Solution

Ok, so just from the problem statement alone, I know we have an object on which two forces are acting; the weight of the object due to gravity and the drag force on the object.

Fnet = mg - KV

Since g is always a constant, it's the acceleration of the drag force we are trying to solve for, correct? Since F=KV, and F=ma, by association we have ma=KV, or a=KV/m, so now we have a function of acceleration in terms of velocity. If we plug this into a=dV/dt, we end up with:

KV/m = dV/dt

or

dt = (m/KV)dV

Integrating both sides, we end up with:

[itex]\int[/itex]dt = (m/K)[itex]\int[/itex](1/V)dV or t = (m/K)ln(V)+C

Now if we solve for V in terms of t we get:

ln(V) = (tK/m)-C or V = e^[(tk/m)-C]

So this is as far as I have been able to get with this problem, because when you plug in initial velocity to solve for C, you get 0 = e^(0-C), but e^x is undefined when x = 0.

Am I making a wrong assumption at the beginning of the problem? I've got about 6 pages of scratch work and I feel this is the closest I've gotten to the actual solution. Any advice would be helpful. Thanks in advance.

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