Question

A balloon having mass 'm' is filled with gas and is held in hands of a boy. Then suddenly it get released and gas starts coming out of it with a constant rate.The velocities of the ejected gases is also constant 2 m/s with respect to the balloon.Find out the velocity of the balloon when the mass of gas is reduced to half.(neglect gravity)

• A. ln 2
• B. 2 ln 4
• C. 2 ln 2
• D. none of these

Answer 1

The correct option is C 2 ln 2

Change in momentum=$\frac{d\overrightarrow{p}}{dt}=0$

$\frac{d\left(mv\right)}{dt}=0$

$v\frac{dm}{dt}+m\frac{dv}{dt}=0$

${\int }_M^{M/2}\frac{dm}{m}={\int }_0^M\frac{dv}{v}$

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

$u=-v\ln \frac{1}{2}$

$u=v\ln 2u=2\ln 2$