AIIMS 2018 Physics Rocket Propulsion MCQ Question

To find the final velocity of the rocket after all the fuel is expelled, we can apply the Tsiolkovsky rocket equation, which states that $\Delta v = v_e \ln\left(\frac{m_i}{m_f}\right)$, where $v_e$ is the exhaust velocity, $m_i$ is the initial mass, and $m_f$ is the final mass. Here, the initial mass $m_i = 5 \times 10^6$ kg, the fuel mass is $4 \times 10^6$ kg, making the final mass $m_f = 1 \times 10^6$ kg, and the exhaust velocity $v_e = 4000$ m/s. Plugging these values into the equation gives $\Delta v = 4000 \ln\left(\frac{5 \times 10^6}{1 \times 10^6}\right) = 4000 \ln(5) \approx 6438$ m/s, confirming that option A is correct.

Options B, C, and D are incorrect as they do not match the calculated velocity based on the logarithmic relationship defined by the rocket equation, indicating a misunderstanding of the mass ratio or the exhaust velocity's impact on the rocket's final speed.