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Question #42279

Difficulty: 8/10 | 🟢 Free

The Question

A particle of mass $m$ is projected radially outwards from the surface of a planet of mass $M$ and radius $R$ . The planet is surrounded by an atmosphere that exerts a drag force on the particle given by $F_d = -\frac{\alpha m}{r^2} v^2$ , where $v$ is the instantaneous velocity of the particle, $r$ is its distance from the center of the planet, and $\alpha$ is a positive constant with dimensions of length. What is the minimum initial velocity $v_0$ required for the particle to escape the planet's gravitational pull?

Options

A.
$\sqrt{\frac{GM}{\alpha} \left( e^{\frac{2\alpha}{R}} - 1 \right)}$
B.
$\sqrt{\frac{2GM}{R} + \frac{GM}{\alpha} \left( 1 - e^{-\frac{2\alpha}{R}} \right)}$
C.
$\sqrt{\frac{2GM}{R} e^{\frac{2\alpha}{R}}}$
D.
$\sqrt{\frac{GM}{\alpha} \left( 1 - e^{-\frac{2\alpha}{R}} \right)}$

Check Your Answer

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

Escape Velocity Independence|Conservative vs Non-conservative|Integration

Why This Question Is Tricky

Common Mistakes

Students often incorrectly assume that escape velocity problems can always be solved using simple mechanical energy conservation ( $K_i + U_i = 0$ ).

This question introduces a non-conservative, position-and-velocity-dependent drag force, rendering standard energy conservation invalid.

The trap lies in attempting to use a simple work-energy theorem without setting up the proper differential equation of motion.

By forcing the student to write $m v (dv/dr) = F_{net}$ and solve the resulting first-order linear ODE using an integrating factor, it tests the transition from elementary conservation laws to rigorous Newtonian dynamics.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Gravitation - Class 11 Part 1
Concept	Escape Velocity Independence
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free