Retarding Force proportional to the velocity falling particl

[Decypher]

Example problem from a Classical Mechanics book.
Find the displacement and velocity of a particle undergoing vertical motion in a medium having a retarding force proportional to the velocity.

F = m(dv/dt) = -mg-kmv

dv/(kv + g) = -dt

(1/k)ln(kv + g) = -t + c

kv + g = e^(-kt+kc)

v = (dz/dt) = (-g/k) + ((k(Vo) + g)/k)e^(-kt)

I don't see how the e^(-kt + kc) is turning into ((k(Vo) + g)/k
I know a property of exponents allows you to separate the exponents powers when they are being added, then each one becomes the power of an exponent which is where e^-kt comes from, but I am not sure about where the other half of the exponent goes.

 

[Buzz Bloom]

F = m(dv/dt) = -mg-kmv

Hi Decypher:

I think y ou have a sigh error. mg is the force of gravity pulling the mass towards rthe Earth, and kmv is a retarding force, slowing down the pull. The direction of these forces are opposite.

Hope this is helpful.

Regards,
Buzz

 

[Decypher]

I spent a great amount of time on this part you are mentioning. The book does not provide a great description on this statement.
"Where -kmv represents a positive upward force since we take z and v = z(dot) to be positive upward, and the motion is downward-that is, v<0, so that -kmv > 0."
I think they are saying that because we are treating a positive velocity as one that is going higher on the z axis, whe have to treat -kmv as kmv because when a negative velocity is plugged in, kmv becomes an opposing force once again. It makes more sense to me to have it be g-kv just like a force opposing gravity should be.

 

[Buzz Bloom]

Find the displacement and velocity of a particle undergoing vertical motion

we are treating a positive velocity as one that is going higher on the z axis

Hi Decypher:

OK. I think the first quoted sentence would have been clearer if the word "upward" had preceded "vertical".

I don't see how the e^(-kt + kc) is turning into ((k(Vo) + g)/k

The above quote is not what you want to do. I suggest you take the equation

kv + g = e^(-kt+kc)

and fill in the missing step: subtract g from both sides, and divide both sides by k. Then make the substitution v = dz/dt.

Then you should be able to make a substitution for c (which is an arbitrary constant of integration) which will give you what you want.

Hope this helps.

Regards,
Buzz