Question

If dv/dt = 6 - 3 v then how velocity will become constant after a long time

Answer 1

$\frac{dv}{6-3v}=dt$

Integrate the above equation within the limits $v_{o}$ to $v$ for velocity and 0 to $t$ for time, we get,
$\int\limits^v_{v_{o}} { \frac{1}{6-3v} } \, dv = \int\limits^0_t \, dt$
㏑($\frac{6-3v}{6-3v_{0}}$)=$-3t$
On solving the above equation we get,
$v=(v_{0}-2) e^{-3t} +2$
As t→ ∞, $e^{-3t}$ becomes 1
Hence, $v=v_{0}$