Rocket Propulsion - Practice Questions & MCQ

Let us assume a rocket of total initial mass (rocket + fuel) m₀, starts moving upward due to thrust force of fuel jet. Assuming the velocity of fuel jet with respect to the rocket to be u (assumed to be constant for this discussion) in vertically downward direction and mass of jet fuel emerging out of rocket per unit time to be $\frac{d m}{d t}$. Let the velocity of rocket after t time of motion be v and the acceleration of the rocket be a in vertically upward direction.

1. Thrust on the rocket

$
F=-\frac{u d m}{d t}
$

Where F= Thrust

$
\frac{d m}{d t}=\text { rate of ejection of the fuel }
$

$u=v e l o c i t y$ of exhaust gas with respect to rocket
$m=$ mass of the rocket at time $t$
Net force on rocket-

$
\begin{aligned}
& F_{n e t}=-\frac{u d m}{d t}-m g \\
& * F=-\frac{u d m}{d t}[\text { if gravity neglected }]
\end{aligned}
$

2. Acceleration of Rocket (a)

$
a=-\frac{u}{m} \frac{d m}{d t}-g
$

- If g is neglected then

$
a=-\frac{u}{m} \frac{d m}{d t}
$

3. Instantaneous Velocity of Rocket (v)

 If g is neglected then-

$
\begin{aligned}
& a=-\frac{u}{m} \frac{d m}{d t} \\
& \frac{d v}{d t}=-\frac{u}{m} \frac{d m}{d t} \\
& \int_0^v d v=-u \int_{m_0}^m \frac{1}{m} d m \\
& \Rightarrow v=u \log _e\left(\frac{m_{\circ}}{m}\right) \\
& v=u * \log _e\left(\frac{m_{\circ}}{m}\right)=2.303 u * \log _{10}\left(\frac{m_{\circ}}{m}\right)
\end{aligned}
$

4. Burnt speed of Rocket
- It is the speed attained by the rocket when complete fuel gets burnt.
- It is the maximum speed attained by the rocket
- Formula

$
\begin{aligned}
V_b & =V_{\max }=u \log _e\left(\frac{m_{\circ}}{m_r}\right) \\
V_b & \rightarrow \text { burnt speed }
\end{aligned}
$

$m_r \rightarrow$ residual mass of empty container