A rocket of initial mass M 0 (including fuel) is moving in free space with velocity V 0 i ^ . It ejects fuel at a constant rate k (mass per unit time) with a constant relative speed v r with res

Question

A rocket of initial mass M 0 ​ (including fuel) is moving in free space with velocity V 0 ​ i ^ . It ejects fuel at a constant rate k (mass per unit time) with a constant relative speed v r ​ with respect to the rocket, in the − i ^ direction. If the mass of the rocket after time t is M ( t ) , what is the velocity of the rocket at time t ?

TestHub · Accepted Answer

The rocket equation for velocity change is $d V = - v_{r} \frac{d M}{M}$ . Integrating this from initial velocity $V_{0}$ to $V (t)$ and initial mass $M_{0}$ to $M (t)$ : $\int_{V_{0}}^{V (t)} d V = - v_{r} \int_{M_{0}}^{M (t)} \frac{d M}{M}$ $V (t) - V_{0} = - v_{r} [ln M]_{M_{0}}^{M (t)} = - v_{r} (ln M (t) - ln M_{0}) = v_{r} (ln M_{0} - ln M (t))$ $V (t) - V_{0} = v_{r} ln (\frac{M _{0}}{M ( t )})$

Since the exhaust is in the $- \hat{i}$ direction and the initial velocity is $V_{0} \hat{i}$ , the velocity gain will be in the $+ \hat{i}$ direction. So, $V (t) = V_{0} \hat{i} + v_{r} ln (\frac{M _{0}}{M ( t )}) \hat{i}$ .

Physics - Center of Mass Question with Solution | TestHub

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JEE Main 2028 (Class 11)