Question

The motion of a body falling from rest in a medium is given by equation dv/dt=1-2v.The velocity at ant time?

Answer 1

Given that the motion of a body falling from rest in a medium is

$\frac{dv}{dt} =1-2v$. Solving this differential equation,

$\frac{dv}{dt} =1-2v\\ \frac{dv}{1-2v} =dt\\ \int \frac{dv}{1-2v} =\int dt\\ -\frac{1}{2} \ln |1-2v|=t+C$

At $t=0$, $v=0$. Hence,

$-\frac{1}{2} \ln |1-0|=0+C \\ C=0$.

The velocity as a function of time is

$-\frac{1}{2} \ln |1-2v|=t\\ \ln |1-2v|=-2t\\ 1-2v=e^{-2t}\\ v=\frac{1}{2}(1-e^{-2t})$