Topic: Propulsion / Aerospace
A rocket with an initial mass of 10,000 kg ejects mass at a constant rate of 100 kg/s and at constant velocity through the nozzle. What should be the exit jet velocity, so as to increase the velocity of the rocket by 2 km/s in 50 seconds. [Assume no drag and optimal expansion in nozzle. Consider, $e^{0.693}\simeq 2$]
- A. $2886~m/s$
- B. $3286~m/s$
- C. $1000~m/s$
- D. $1985~m/s$
Answer: A. $2886~m/s$
Explanation: Using the ideal rocket equation (Tsiolkovsky equation): $\Delta V = v_e \ln\left(\frac{m_i}{m_f}\right)$.
Initial mass, $m_i = 10000$ kg.
Mass flow rate, $\dot{m} = 100$ kg/s.
Time, $t = 50$ s.
Mass ejected = $\dot{m} \times t = 100 \times 50 = 5000$ kg.
Final mass, $m_f = 10000 - 5000 = 5000$ kg.
Target $\Delta V = 2$ km/s = 2000 m/s.
$2000 = v_e \ln\left(\frac{10000}{5000}\right) = v_e \ln(2)$.
Given $e^{0.693} \simeq 2$, we have $\ln(2) = 0.693$.
$2000 = v_e \times 0.693 \implies v_e = \frac{2000}{0.693} = 2886.002~m/s$.