Question

The motion of a body is given by the equation $\frac{d v}{d t}=6.0-3 \,v$, where $v$ is the speed (in $m \,s ^{-1}$ ) and $t$ is time in second. If the body was at rest at $t=0$, then find its speed (in $m / s$ ) when the acceleration is half the initial value.

Answer 1

$a=\frac{d v}{d t}=(6-3 v)$
$\int \frac{d v}{6-3 v}=\int d t$
$\left.\frac{\ln (6-3 v)}{-3}\right|_{0} ^{v}=t$
$-\frac{1}{3}\left[\ln \left\{\frac{(6-3 v)}{(6)}\right\}\right]=t$
$-\frac{1}{3}\left[\ln \left\{\frac{(2-v)}{(2)}\right\}\right]=t$
$v=2\left(1-e^{-3 t}\right)$
$a=\frac{d v}{d t}=6 e^{-3 t}$
$3=6 e^{-3 t}$
$e^{-3 t}=\frac{1}{2}$
$v=2\left[1-\frac{1}{2}\right]=1 \,m / s$