Q.34 Physics ⭐ 360° Option Elimination Active

NEET 2021 Physics Question 34 Solution

Topic: Gravitation · Official NTA Booklet Code M1

Question

The escape velocity from the Earth's surface is v. The escape velocity from the surface of another planet having a radius, four times that of Earth and same mass density is:

✓

Verified Against NTA Final Answer Key

Option (4) : 4v

📚 Gravitation Easy

📋 Given Parameters & Problem Setup

Escape Velocity on Earth v

New Planet Radius (R') 4 R_E

New Planet Density (ρ') Equal to Earth (same mass density ρ)

⚡

The 30-Second Exam Hall Shortcut: ⚡ Constant Density Escape Velocity Shortcut: v_e ∝ R

⏱️ Save ~20 seconds

$v_e = \sqrt{\frac{2GM}{R}} = \sqrt{\frac{2G(\frac{4}{3}\pi R^3 \rho)}{R}} = R \sqrt{\frac{8}{3}\pi G \rho} \implies v_e \propto R$ ! Since the planet's radius is 4 times that of Earth at identical density, the escape velocity is directly $4v$ ! Instant 5-second answer matching Option (4)!

1 Express mass in terms of density:

M = \frac{4}{3}\pi R^3 \rho

2 Substitute into escape velocity formula:

v_e = \sqrt{\frac{2G M}{R}} = \sqrt{\frac{2G}{R} \left(\frac{4}{3}\pi R^3 \rho\right)} = R \sqrt{\frac{8}{3}\pi G \rho}

3 For equal mass density (

\rho = \text{constant}

v_e \propto R

4 Since

R' = 4 R

, we have

v' = 4 v

5 Match Option (4).

🛡️

Why You Can 100% Trust This Shortcut: Canonical Newton's law of gravitation and spherical planetary mass-density integration.

📐

Step-by-Step Rigorous Workflow & Mathematical Proof

Step 1

Standard Formula for Escape Velocity

The minimum velocity required for a particle to escape gravitational attraction from the surface of a spherical planet of mass $M$ and radius $R$ is:

v_e = \sqrt{\frac{2GM}{R}}

Step 2

Expressing Escape Velocity in Terms of Density

Assuming uniform mass density $\rho$ , the planetary mass is $M = \frac{4}{3}\pi R^3 \rho$ . Substituting into the escape velocity formula gives:

v_e = \sqrt{\frac{2G \left(\frac{4}{3}\pi R^3 \rho\right)}{R}} = \sqrt{\frac{8}{3}\pi G \rho R^2} = R \sqrt{\frac{8\pi G \rho}{3}}

Step 3

Comparing the New Planet with Earth

For two planets of identical density $\rho$ , escape velocity is directly proportional to planetary radius:

\frac{v'}{v} = \frac{R'}{R} = \frac{4R}{R} = 4 \implies v' = 4v

⭐

360° Option Elimination Engine Key Feature

Diagnose every choice: Know exactly why the right option works and why distractors fail.

4-Option Deep Scan

1 1

✗ Incorrect

Verdict & Scientific Reason:

Assumes escape velocity is independent of planetary radius.

⚠️

Student Fallacy (Why candidates pick this trap): Believing escape speed is a universal constant.

2 2

✗ Incorrect

Verdict & Scientific Reason:

Result of applying $v_e \propto \sqrt{R}$ , which only applies if planet MASS is held constant ( $M = \text{const}$ ), not density.

⚠️

Student Fallacy (Why candidates pick this trap): Using

v_e = \sqrt{2GM/R}

with constant

M

instead of constant

\rho

3 3

✗ Incorrect

Verdict & Scientific Reason:

Unphysical integer guess.

⚠️

Student Fallacy (Why candidates pick this trap): Random estimation.

4 4

✓ 100% Correct

Verdict & Scientific Reason:

Exact escape velocity for a planet with $4\times$ radius and identical density ( $v_e \propto R$ ).

💡

Why this is airtight: None; rigorous gravitational density scaling.

⚠️

NTA Trap Alert: Constant Mass vs. Constant Density Proportionality Trap

If planetary MASS is constant: $v_e \propto \frac{1}{\sqrt{R}}$ . If planetary DENSITY is constant: $M \propto R^3$ , making $v_e \propto R$ . Candidates who mistakenly use constant mass obtain $v/2$ or $2v$ (Option 2).

🛡️ How to avoid the -1 mark: Identify the invariant: "same mass density"

\implies v_e \propto R

. Since

R

4\times

v

4\times

📖

Core NEET Syllabus Standard

NCERT Physics · Gravitation Standard Benchmark

Core Scientific Principle Tested:

"Escape velocity from a celestial body of uniform density $\rho$ and radius $R$ is $v_e = R\sqrt{\frac{8\pi G\rho}{3}}$ , which scales linearly with radius."

Syllabus Focus: Escape Speed

✓ Core NEET UG Syllabus Anchor Syllabus & Academic Standards ↗

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NEET 2021 Physics Question 34 Solution

Question

The 30-Second Exam Hall Shortcut: ⚡ Constant Density Escape Velocity Shortcut: v_e ∝ R

Step-by-Step Rigorous Workflow & Mathematical Proof

Standard Formula for Escape Velocity

Expressing Escape Velocity in Terms of Density

Comparing the New Planet with Earth

360° Option Elimination Engine Key Feature

NTA Trap Alert: Constant Mass vs. Constant Density Proportionality Trap

Core NEET Syllabus Standard

Ready to attempt under real exam pressure?

🔗 Related Questions — Gravitation

Escape Speed Ratio for Equal-Mass Planets with Different Radii

Kepler's Third Law Orbital Period Proportionality to Semi-Major Axis / Radius

Kepler’s Third Law and Orbital Period of Planets

Variation of Weight / Gravitational Force at Finite Height h = R/3

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