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Question #42418

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The Question

A toy rocket of initial mass $m_0$ kg is placed on the smooth horizontal floor of a train. The train accelerates forward with a constant acceleration $a_0 \text{ m/s}^2$ . The rocket is pointed forward. At $t=0$ s, the rocket motor is ignited, ejecting gas backwards at a constant mass rate $\mu \text{ kg/s}$ and at a constant speed $u \text{ m/s}$ relative to the rocket. Given that the constant thrust provided by the motor is $u\mu = \frac{1}{2} m_0 a_0$ , what is the maximum backward speed of the rocket relative to the train in m/s? (Assume the rocket has sufficient fuel and does not hit the back wall of the train).

Options

A.
$\frac{m_0 a_0}{2\mu} (1 - \ln 2)$
B.
$\frac{m_0 a_0}{\mu} (1 - \ln 2)$
C.
$\frac{m_0 a_0}{2\mu} \ln 2$
D.
$\frac{m_0 a_0}{\mu} \left(1 - \frac{1}{e}\right)$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Frame of Reference Inertial vs Non-Inertial|Variable Mass Systems|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Generalized Newton's Second Law for a variable mass system in a non-inertial frame.

The critical trap lies in the formulation of the pseudo-force.

Students frequently assume the pseudo-force is constant ( $-m_0 a_0$ ), forgetting that it must scale with the instantaneous mass $m(t)$ of the system.

By setting the constant thrust to be initially weaker than the pseudo-force, the rocket moves backward.

As mass is ejected, the pseudo-force decreases until it perfectly balances the thrust, marking the point of maximum backward speed.

Failing to couple the mass depletion with the non-inertial pseudo-force leads to an incorrect continuously accelerating backward motion.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Laws of Motion - Class 11 Part 1
Concept	Frame of Reference Inertial vs Non-Inertial
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free