Q.42 Physics ⭐ 360° Option Elimination Active

NEET 2021 Physics Question 42 Solution

Topic: Gravitation · Official NTA Booklet Code M1

Question

A particle of mass 'm' is projected with a velocity v = kVe (k < 1) from the surface of the earth. (Ve = escape velocity) The maximum height above the surface reached by the particle is:

✓

Verified Against NTA Final Answer Key

Option (4) : R k^2 / (1 - k^2)

📚 Gravitation Medium

📋 Given Parameters & Problem Setup

Projection Speed (v) k V_e with k < 1

Escape Speed (V_e) √(2 G M / R)

⚡

The 30-Second Exam Hall Shortcut: ⚡ Height from Escape Fraction: h = R k² / (1 - k²)

⏱️ Save ~25 seconds

Conservation of mechanical energy with $v = k V_e$ : $\frac{1}{2} m (k^2 V_e^2) - \frac{GMm}{R} = -\frac{GMm}{R + h}$ . Since $\frac{1}{2} V_e^2 = \frac{GM}{R}$ , this simplifies directly to $\frac{1}{R+h} = \frac{1-k^2}{R} \implies R+h = \frac{R}{1-k^2} \implies h = \frac{R k^2}{1 - k^2}$ ! Instant 15-second algebraic solution matching Option (4)!

1 Conserve energy:

E_i = \frac{1}{2} m v^2 - \frac{GMm}{R}

at surface;

E_f = -\frac{GMm}{R + h}

at peak (

v = 0

2 Use

V_e^2 = \frac{2GM}{R} \implies \frac{1}{2} m v^2 = \frac{1}{2} m k^2 \left(\frac{2GM}{R}\right) = \frac{GMm}{R} k^2

3 Substitute:

\frac{GMm}{R} k^2 - \frac{GMm}{R} = -\frac{GMm}{R + h}

4 Divide by

-GMm

\frac{1 - k^2}{R} = \frac{1}{R + h}

5 Invert:

R + h = \frac{R}{1 - k^2} \implies h = \frac{R}{1 - k^2} - R = \frac{R k^2}{1 - k^2}

6 Match Option (4).

🛡️

Why You Can 100% Trust This Shortcut: Canonical conservation of mechanical energy in inverse-square central gravitational field.

📐

Step-by-Step Rigorous Workflow & Mathematical Proof

Step 1

Initial Energy at the Surface of the Earth

The particle of mass $m$ is projected with speed $v = k V_e$ . The total mechanical energy at the Earth's surface ( $r = R$ ) is:

E_i = \frac{1}{2} m v^2 - \frac{GMm}{R} = \frac{1}{2} m (k V_e)^2 - \frac{GMm}{R}

Step 2

Expressing in Terms of Gravitational Potential Energy

Using the escape velocity expression $V_e^2 = \frac{2GM}{R}$ :

\frac{1}{2} m (k^2 V_e^2) = \frac{1}{2} m k^2 \left(\frac{2GM}{R}\right) = \frac{GMm}{R} k^2

Step 3

Equating with Final Energy at Maximum Height

At maximum height $h$ above the Earth's surface, the radial distance is $r = R + h$ and velocity drops momentarily to zero ( $K_f = 0$ ):

E_f = -\frac{GMm}{R + h}

Step 4

Solving for Maximum Height h

By conservation of mechanical energy $E_i = E_f$ :

\frac{GMm}{R} k^2 - \frac{GMm}{R} = -\frac{GMm}{R + h} \implies -\frac{GMm}{R}(1 - k^2) = -\frac{GMm}{R + h} \implies \frac{1}{R + h} = \frac{1 - k^2}{R}

Step 5

Final Algebraic Simplification

Inverting both sides and isolating $h$ :

R + h = \frac{R}{1 - k^2} \implies h = \frac{R}{1 - k^2} - R = R\left(\frac{1 - (1 - k^2)}{1 - k^2}\right) = \frac{R k^2}{1 - k^2}

⭐

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:

Has $(1-k)^2$ in denominator instead of $(1-k^2)$ .

⚠️

Student Fallacy (Why candidates pick this trap): Squaring the entire binomial

(1-k)

instead of difference of squares

(1-k^2)

2 2

✗ Incorrect

Verdict & Scientific Reason:

Incorrect binomial form in denominator.

⚠️

Student Fallacy (Why candidates pick this trap): Sign error in energy balance.

3 3

✗ Incorrect

Verdict & Scientific Reason:

Dimensionally inconsistent (has dimension $[L^2]$ instead of $[L]$ ).

⚠️

Student Fallacy (Why candidates pick this trap): Failing to check dimensions of length.

4 4

✓ 100% Correct

Verdict & Scientific Reason:

Exact maximum height derived from gravitational energy conservation.

💡

Why this is airtight: None; flawless energy derivation.

⚠️

NTA Trap Alert: Flat Earth Approximation Trap

Using the uniform gravity approximation $h = v^2 / 2g$ gives $h = k^2 R$ . This is only valid for small heights ( $h \ll R$ ). For speeds approaching escape velocity, the variation of gravity with height $U(r) = -GMm/r$ MUST be used.

🛡️ How to avoid the -1 mark: When speed is given as a fraction of escape velocity, ALWAYS use

-GMm/r

📖

Core NEET Syllabus Standard

NCERT Physics · Gravitation Standard Benchmark

Core Scientific Principle Tested:

"In non-uniform gravitational fields, mechanical energy is conserved: $\frac{1}{2}mv^2 - \frac{GMm}{R} = -\frac{GMm}{R+h}$ ."

Syllabus Focus: Gravitational Potential Energy and Escape Speed

✓ Core NEET UG Syllabus Anchor Syllabus & Academic Standards ↗

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NEET 2021 Physics Question 42 Solution

Question

The 30-Second Exam Hall Shortcut: ⚡ Height from Escape Fraction: h = R k² / (1 - k²)

Step-by-Step Rigorous Workflow & Mathematical Proof

Initial Energy at the Surface of the Earth

Expressing in Terms of Gravitational Potential Energy

Equating with Final Energy at Maximum Height

Solving for Maximum Height h

Final Algebraic Simplification

360° Option Elimination Engine Key Feature

NTA Trap Alert: Flat Earth Approximation Trap

Core NEET Syllabus Standard

Ready to attempt under real exam pressure?

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