m(dv/dt)=mg−kv2
Put mg=ku02 and u0=gT. Then
m(dv/dt)=k(u02−v2)
1. Consider the case u0>v0
u02−v2dv=mkdt
v(0)∫v(t)u02−v2dv=0∫tmkdt=mkt
v(0)∫v(t)u02−v2dv=v0∫v(t)(u0−v1+u0+v1)2u0dv=
=2u01logu0−v(t)u0+v(t)−2u01logu0−v0u0+v0
logu0−v(t)u0+v(t)=logu0−v0u0+v0+2tmku0
But mku0=gmgku0=ku02gku0=u0g=T1
u0−v(t)u0+v(t)=u0−v0u0+v0em2kv0t=u0−v0u0+v0eT2t=eT2(t−t0) ,
where t0=−21logu0−v0u0+v0 is the time when v(t)=0. Then
u0−v(t)2u0=1+u0−v(t)u0+v(t)=1+eT2(t−t0)
u0−v(t)=1+eT2(t−t0)2u0
v(t)=u0−1+eT2(t−t0)2u0=u0eT2(t−t0)+1eT2(t−t0)−1=u0tanhTt−t0
where u0=kmg, T=gu0=kgm and t0=−21logu0−v0u0+v0.
(b) t→+∞limv(t)=t→+∞limu0tanhTt−t0=u0=kmg
(c) s(t)=s(0)+0∫tv(t)dt=0∫tu0tanhTt−t0dt=
=u0T0∫tcoshTt−t0dcoshTt−t0=u0T(logcoshTt−t0−logcoshTt0)
=u0TlogcoshTt0coshTt−t0
2. Consider the case u0<v0
u02−v2dv=mkdt
v(0)∫v(t)u02−v2dv=0∫tmkdt=mkt
v(0)∫v(t)u02−v2dv=v0∫v(t)(u0−v1+u0+v1)2u0dv=
=2u01logv(t)−u0v(t)+u0−2u01logv0−u0v0+u0
logv(t)−u0v(t)+u0=logv0−u0v0+u0+2tmku0=logv0−u0v0+u0+T2t
v(t)−u0v(t)+u0=v0−u0v0+u0eT2t=eT2(t−t0) ,
where t0=−21logv0−u0v0+u0 is the time when v(t)=∞. Then
v(t)−u02u0=v(t)−u0v(t)+u0−1=eT2(t−t0)−1
v(t)−u0=eT2(t−t0)−12u0
v(t)=u0+eT2(t−t0)−12u0=u0eT2(t−t0)−1eT2(t−t0)+1=u0cothTt−t0
where u0=kmg, T=gu0=kgm and t0=−21logv0−u0v0+u0.
3. Consider the case v0=u0. It is evident that in this case v(t)=u0 is the unique solution.
(b) t→+∞limv(t)=t→+∞limu0tanhTt−t0=u0 , if u0>v0 ,
t→+∞limv(t)=t→+∞limu0cothTt−t0=u0 , if u0<v0 ,
t→+∞limv(t)=t→+∞limu0=u0 , if u0=v0 .
Therefore, in any case t→+∞limv(t)=u0=kmg.
(c) s(t)=s(0)+0∫tv(t)dt=0∫tu0tanhTt−t0dt=
=u0T0∫tcoshTt−t0dcoshTt−t0=u0T(logcoshTt−t0−logcoshTt0)
=u0TlogcoshTt0coshTt−t0 , if u0>v0
If u0<v0 then s(t)=s(0)+0∫tv(t)dt=0∫tu0cothTt−t0dt=
=u0T0∫tsinhTt−t0dsinhTt−t0=u0T(logsinhTt−t0−logsinhTt0)
=u0TlogsinhTt0sinhTt−t0
If u0=v0 then s(t)=s(0)+0∫tv(t)dt=0∫tu0dt=u0t
Answer.
if u0>v0 then
(a) v(t)=u0tanhTt−t0, where u0=kmg, T=gu0=kgm and t0=−21logu0−v0u0+v0
(b) t→+∞limv(t)=kmg
(c) s(t)=u0TlogcoshTt0coshTt−t0
if u0<v0 then
(a) v(t)=u0cothTt−t0, where u0=kmg, T=gu0=kgm and t0=−21logv0−u0v0+u0
(b) t→+∞limv(t)=kmg
(c) s(t)=u0TlogsinhTt0sinhTt−t0
if u0=v0 then
(a) v(t)=v0=u0
(b) t→+∞limv(t)=kmg ,
(c) s(t)=u0t
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