Question

The velocity of a spherical ball through a viscous liquid is given by $v=v_0(1-e^{kt})$, where $v_0$ is the initial velocity and $t$ represents time. If $k$ depends on radius of ball ($r$), coefficient of viscosity $\left(\eta \right)$ and mass of the ball ($m$), then

• A. $k=mr/\eta$
• B. $k=\eta m/r$
• C. $k=r\eta /m$
• D. $k=mr$

Answer 1

The correct option is C $k=r\eta /m$

The velocity of the ball is given by:

$v=v_0(1-e^{kt})$

The value of the $k$ depends on the following factors:
$k\propto r^{\alpha }{\eta }^{\beta }{m}^{\gamma }$

$kt=M^0{L}^0{T}^0\Rightarrow K=T^{-1}$

$T^{-1}\alpha {\left(L^1\right)}^{\alpha }{\left(M{L}^{-1}{T}^{-1}\right)}^{\beta }{\left(M^1\right)}^{\gamma }$
$\Rightarrow \alpha =1,\beta =1,\gamma =-1$