Question

What will be the mass of the rocket at a certain instant of time $t$, if it moves with a constant acceleration $a$, in absence of external forces?

The gas escapes with a constant velocity $u$ relative to the rocket, and initial total mass of the rocket was $m_0$.

• A. $m_0{e}^{-at/u}$
• B. $m_0{e}^{-at/2u}$
• C. $m_0{e}^{at/2u}$
• D. $m_0{e}^{at/u}$

Answer 1

The correct option is A $m_0{e}^{-at/u}$

Using,

$F_{\text{net}}=-v_{\text{rel}}\left(\frac{dm}{dt}\right)=-u\left(\frac{dm}{dt}\right)=ma$

$\Rightarrow {\int }_{m_0}^m\frac{dm}{m}={\int }_0^t\frac{-adt}{u}$

$\Rightarrow {\log }_e{\left[m\right]}_{m_0}^m=\frac{-at}{u}$

$\Rightarrow m=m_0{e}^{-at/u}$

Hence, option $\left(A\right)$ is the correct answer.