GATE 2023 AE – Question 38

Propulsion · Rockets: thrust equation, specific impulse, multi-staging, solid and liquid propellant rockets · 2 marks · Multiple choice

Two missiles have equal specific impulse and initial mass 5600 kg. A burns 120 kg/s for 30 s; B burns 70 kg/s for 70 s. Neglect gravity and drag. Their final speeds satisfy

$V_A=4.1V_B$
$V_A=V_B$
$V_A=0.5V_B$
$V_A=0.7V_B$

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Correct answer: (C) $V_A=0.5V_B$

Explanation

**Rocket equation** (no gravity or drag): the final speed is
$$V=I_{sp}\,g_0\ln\frac{m_0}{m_f},$$
so for equal specific impulse the speed depends only on $\ln(m_0/m_f)$.

**Missile A:** burns $120\times30=3600$ kg, so $m_f=5600-3600=2000$ kg.
$$\ln\frac{5600}{2000}=\ln2.8=1.030 .$$

**Missile B:** burns $70\times70=4900$ kg, so $m_f=5600-4900=700$ kg.
$$\ln\frac{5600}{700}=\ln8=2.079 .$$

$$\frac{V_A}{V_B}=\frac{1.030}{2.079}=0.495\approx0.5 .$$

So $V_A=0.5\,V_B$ (option C).