
(a) Block B will come to rest, it force applied to it will vanish due to frictional force acting between block B and surface $ie,$
$force\ applied = frictional\ force$
$ie,\ \ \mu\ mg = ma$
$or\ \ \ \ \ \ \ \ \ \mu\ mg = m\left( \frac{v}{t} \right)$
$or\ \ \ \ \ t = \frac{v}{\mu\ g}$