Question

A balloon having mass ${}^{\prime }{m}^{\prime }$ is filled with gas and is held in hands of a boy. Then suddenly it get released and gas stars coming out of it with a constant rate. The velocities of the ejected gases is also constant $2m/s$ with respect to the balloon. Find the velocity of the balloon when the mass is reduced to half.

• A. $ln2$
• B. $2ln4$
• C. $2ln2$
• D. none of these

Answer 1

Answer: (C)

$2ln2$

Force is given as the rate of change of momentum with respect to time.
$F_{ext}=\frac{dp}{dt}=\frac{d}{dt}\left(mv\right)=v\frac{dm}{dt}+m\frac{dv}{dt}$

There is no external force acting on the balloon. Therefore, $F_{ext}=0$

Hence, we have,
$v_{rel}\frac{dm}{dt}=-m\frac{dv}{dt}$

or, $v_{rel}dm=-mdv$

Intergrating,

${\int }_m^{\frac{m}{2}}\frac{dm}{m}=-\frac{1}{2}{\int }_0^{v}dv$

or, ${\left[lnm\right]}_m^{\frac{m}{2}}=-\frac{1}{2}$

or, $ln\left(\frac{\frac{m}{2}}{m}\right)=-\frac{1}{2}v$

or, $v=-2ln\frac{1}{2}$

or, $v=2ln2$ is our required answer.
Therefore the correct answer is option (C).