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On a body the forces are operated: P  mg – weight of
2
 ds 
2
body (in the direction of motion) and F  kv   k  –
 dt 
(against direction of motion), where k – coefficient of
proportion.
After the second law of Newton we have:

mw  P  F
or
d 2 s  ds  2
m  mg   k  .
dt 2  dt 

We have differential equation of the second order, which
does not contain an unknown function (ts ) obviously. Pursuant
to problem specification we will write down such initial
conditions:
ds ) 0 (
. ) 0 ( s  , 0  ) 0 ( v  0
dt
ds d 2 s dv
We will lay down  v then  and will obtain
dt dt 2 dt
equation of the first order with the separated variables
dv 2
m  mg  kv
dt
or
m dv 2 2 2 mg
 a  v , where a  .
k dt k

Separating variables and integrating, we find

dv k 1 a  v k
 dt , ln  t  C .
1
2
a  v 2 m 2a a  v m

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