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Question #42466

Difficulty: 8/10 | 🟢 Free

The Question

A satellite of mass $m$ is in a circular orbit of radius $R$ around a planet of mass $M$ . The satellite fires its thruster, instantaneously ejecting a small mass $\Delta m$ ( $\Delta m \ll m$ ) with an exhaust speed $u$ relative to the satellite. The mass is ejected in a direction exactly perpendicular to the satellite's instantaneous orbital plane. If $v_0 = \sqrt{GM/R}$ is the initial orbital speed, which of the following represents the eccentricity $e$ of the new orbit? (Assume the change in the satellite's mass is negligible for subsequent orbital dynamics).

Options

A.
$\left( \frac{\Delta m \cdot u}{m v_0} \right)^2$
B.
$\frac{\Delta m \cdot u}{m v_0}$
C.
$2 \left( \frac{\Delta m \cdot u}{m v_0} \right)$
D.
$\sqrt{1 + \left( \frac{\Delta m \cdot u}{m v_0} \right)^2}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Recoil and Thrust Forces|Weightlessness in Orbit|Vector Cross-Product

Why This Question Is Tricky

Common Mistakes

This question tests the misconception that an impulse applied perpendicular to an orbital plane merely tilts the plane without altering the orbit's shape.

Students often incorrectly assume $e=0$ or misapply the linear impulse-momentum theorem by assuming a linear relationship between $\Delta v$ and $e$ (which is only true for radial impulses).

By requiring the transition from a non-inertial variable-mass recoil to the perturbed Keplarian state vectors (Vis-viva equation and specific angular momentum cross-product), the problem exposes the non-linear dependence of eccentricity on out-of-plane velocity perturbations.

It rigorously blends Newton's second law for variable mass with 3D orbital dynamics.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Laws of Motion - Class 11 Part 1
Concept	Recoil and Thrust Forces
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free