A rocket of initial mass M (including fuel) is moving in free space with a constant velocity V 0 . It ejects fuel at a constant relative speed v r with respect to the rocket. If the mass of the ro

Question

A rocket of initial mass M (including fuel) is moving in free space with a constant velocity V 0 ​ . It ejects fuel at a constant relative speed v r ​ with respect to the rocket. If the mass of the rocket decreases by a small amount Δ m in a short time interval Δ t , the change in the rocket's velocity is approximately:

TestHub · Accepted Answer

Consider the rocket and ejected fuel as a system. In the absence of external forces, the total momentum of the system is conserved.

Let the mass of the rocket at time $t$ be $M$ and its velocity be $V$ . After time $Δ t$ , the mass of the rocket becomes $M - Δ m$ and its velocity becomes $V + Δ V$ . The ejected fuel has mass $Δ m$ and its velocity relative to the ground is $V - v_{r}$ (since $v_{r}$ is relative speed backwards). Conservation of momentum: $M V = (M - Δ m) (V + Δ V) + Δ m (V - v_{r})$ . $M V = M V + M Δ V - Δ mV - Δ m Δ V + Δ mV - Δ m v_{r}$ . Neglecting the product of small quantities $Δ m Δ V$ : $0 = M Δ V - Δ m v_{r}$ . Therefore, $Δ V = \frac{Δ m}{M} v_{r}$ .

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JEE Main 2027 (Class 12+)