Question

Find the mass $\left(m\right)$ of the rocket as a function of time, if it moves with a constant acceleration $a$ in the absence of external force. The gas escapes with a constant velocity $u$ relative to the rocket and its initial mass was $m_0$.

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

Answer 1

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


Since $F_{ext}=0$, we get
$F_{net}=V_r\left(\frac{-dm}{dt}\right)=u\frac{-dm}{dt}$ ... (I)
Also $F_{net}=ma$ ...(II)

By eq. (I) and (II)
$ma=-u\cdot \frac{dm}{dt}$
Integrating,
${\int }_{m_0}^m\frac{dm}{m}={\int }_0^t\frac{-adt}{u}$
$\Rightarrow \left[\ln \frac{m}{m_0}\right]=\frac{-at}{u}$
$\Rightarrow \frac{m}{m_0}=e^{-at/u}$
$\Rightarrow m=m_0{e}^{-at/u}$