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Home Q & A Board Science-Related Homework Help Physics
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toni toni
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11 years ago
Here is the actual question.

An object of mass m is dropped from a high altitude. How long will it take the object to achieve a velocity equal to one half of its terminal velocity if the drag force is assumed to be proportional to the velocity?

I'm supposed to solve using differential equation. I'm just so lost at this point in time.
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#1
11 years ago
mdv/dt = mg - cv
dv/dt = g - (c/m) v

let k = c/m

dv/dt = g - kv
dv /(g - kv) = dt

(-1/k) ln(g - kv) = t + B
ln(g - kv) = -kt - kB
g - kv = e^(-kt) e^(-kB)
@ t = 0, v = 0
 g - kv = e^(-kt) e^(-kB)
 g - 0 = 1* e^(-kB)

g - kv = ge^(-kt)
kv = g - ge^(-kt)

v = (g/k)[1 - e^(-kt) ]
`````````````````````````````
terminal velocity occurs when dv/dt becomes 0
dv/dt = g - kv
so v =  g/k

half of terminal velocity = (g/2k)

g/(2k) = (g/k)[1 - e^(-kt) ]
1/2 = 1 - e^(-kt)
e^(-kt) = 1/2
-kt = ln(1/2)
t = (1/k) ln2
```````````````
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