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Question #42293

Difficulty: 8/10 | 🟢 Free

The Question

A rocket of initial mass $M_0$ is launched vertically upwards from the Earth's equator. It ejects gas vertically downwards (relative to the rocket) at a constant relative speed $u$ and a constant rate $\mu = \frac{M_0}{\tau}$ . The Earth rotates with a constant angular speed $\omega$ . Neglecting air resistance, the variation of gravity with height, and assuming the lateral drift is small enough that the rocket's orientation remains perfectly vertical relative to the Earth, which of the following expressions gives the magnitude of the lateral drift velocity of the rocket at time $t = \frac{\tau}{2}$ ? (Let $g$ be the true acceleration due to gravity and $R$ be the radius of the Earth).

Options

A.
$\omega \tau \left[ u(1 - \ln 2) - \frac{1}{4}(g - \omega^2 R)\tau \right]$
B.
$\omega \tau \left[ u(1 - \ln 2) - \frac{1}{4}g\tau \right]$
C.
$2\omega \tau \left[ u \ln 2 - \frac{1}{4}(g - \omega^2 R)\tau \right]$
D.
$\frac{1}{2}\omega \tau \left[ u(1 - \ln 2) - \frac{1}{4}(g - \omega^2 R)\tau \right]$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Effect of Earth's Rotation on g|Variable Mass Systems|Relative Velocity in 1D

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of variable mass dynamics and non-inertial reference frames.

A common trap is failing to recognize that the Coriolis force is proportional to the instantaneous mass, meaning the lateral acceleration $a_x = 2\omega v_z$ is independent of mass, whereas the vertical velocity $v_z$ is highly dependent on the logarithmic mass decay.

Furthermore, students often blindly use $g$ instead of the effective gravity $g_{eff} = g - \omega^2 R$ at the equator, ignoring the centrifugal contribution in the rotating frame.

Finally, the integration of the logarithmic velocity profile tests mathematical rigor under time constraints.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Gravitation - Class 11 Part 1
Concept	Effect of Earth's Rotation on g
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free