The motion of a body falling from rest in a medium is given by the equation #d/dt(v)#=1-2#v#. What is the velocity at any time?Express in terms of time only.

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Answer 1 · 2018-04-06T17:04:56

The answer is #v=1/2(1-e^(-2t))#

The problem is solving a first order diffential equation

#(dv)/dt=1-2v#

#(dv)/(1-2v)=dt#

#int(dv)/(1-2v)=intdt#

#-1/2ln(1-2v)=t+C#

The initial conditions are

#v=0# when #t=0#

#-1/2ln(1)=0+C#

#C=0#

Therefore,

#-1/2ln(1-2v)=t#

#ln(1-2v)=-2t#

#1-2v=e^(-2t)#

#2v=1-e^(-2t)#

#v=1/2(1-e^(-2t))#