Q.46 Physics · Section B ⭐ 360° Option Elimination Active

NEET 2024 Physics Question 46 Solution

Topic: Gravitation · Official NTA Booklet Code T1

Question

The minimum energy required to launch a satellite of mass m from the surface of earth of mass M and radius R in a circular orbit at an altitude of 2R from the surface of the earth is:

✓

Verified Against NTA Final Answer Key

Option (3) : 5 GmM / (6R)

📚 Gravitation Medium

📋 Given Parameters & Problem Setup

Mass of satellite m

Mass of Earth M

Radius of Earth R

Altitude of circular orbit h = 2R (orbital radius r = R + 2R = 3R)

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The 30-Second Exam Hall Shortcut: ⚡ Total Energy Balance Shortcut

⏱️ Save ~45 seconds

E_launch = E_final - U_initial = (-GmM / 2r) - (-GmM / R) = GmM/R × (1 - 1/2(3)) = GmM/R × (1 - 1/6) = 5GmM / 6R!

1 Orbital radius r = R + h = R + 2R = 3R.

2 Total mechanical energy of satellite in circular orbit: E_final = -GmM / (2r) = -GmM / (6R).

3 Initial mechanical energy on Earth’s surface (at rest before launch): E_initial = U_initial = -GmM / R.

4 Minimum launch energy required: E_launch = E_final - E_initial.

5 E_launch = -GmM / (6R) - (-GmM / R) = GmM/R × (1 - 1/6) = 5GmM / (6R).

6 Match Option (3) in under 15 seconds!

🛡️

Why You Can 100% Trust This Shortcut: Conservation of Mechanical Energy in Gravitational Field (E_launch = E_orbital - U_surface).

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Step-by-Step Rigorous Workflow & Mathematical Proof

Step 1

Initial Energy of the Satellite on Earth’s Surface

On the surface of the Earth, before launch, the satellite is at rest (K_i = 0) at radial distance r_i = R from the center of the Earth. Its initial mechanical energy is entirely gravitational potential energy:

E_i = K_i + U_i = 0 + \left(-\frac{G M m}{R}\right) = -\frac{G M m}{R}

Step 2

Orbital Radius Determination

The satellite is to be placed in a circular orbit at an altitude h = 2R above the surface of the Earth. The orbital radius r measured from the center of the Earth is:

r = R + h = R + 2R = 3R

Step 3

Total Mechanical Energy in Circular Orbit

A satellite of mass m in a circular orbit of radius r has orbital speed v_o = √(GM/r), kinetic energy K = GmM / (2r), and potential energy U = -GmM / r. The total mechanical energy is:

E_f = K_f + U_f = \frac{G M m}{2r} - \frac{G M m}{r} = -\frac{G M m}{2r} = -\frac{G M m}{2(3R)} = -\frac{G M m}{6R}

Step 4

Calculation of Minimum Launch Energy

By the work-energy theorem, the minimum energy E_launch that must be supplied to launch the satellite from the surface into orbit is the difference between final and initial total mechanical energies:

E_{launch} = E_f - E_i = -\frac{G M m}{6R} - \left(-\frac{G M m}{R}\right) = \frac{G M m}{R} \left(1 - \frac{1}{6}\right) = \frac{5 G M m}{6 R}

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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:

Represents the kinetic energy needed just to orbit at the surface (r = R).

⚠️

Student Fallacy (Why candidates pick this trap): Ignoring altitude 2R.

2 (2)

✗ Incorrect

Verdict & Scientific Reason:

Incorrect algebra.

⚠️

Student Fallacy (Why candidates pick this trap): Confusing potential energy with total energy.

3 (3)

✓ 100% Correct

Verdict & Scientific Reason:

Exact energy balance: -GmM/(6R) - (-GmM/R) = 5GmM/(6R).

💡

Why this is airtight: None.

4 (4)

✗ Incorrect

Verdict & Scientific Reason:

Represents only the change in potential energy ΔU = GmM/R - GmM/3R = 2GmM/3R, completely forgetting the kinetic energy required to orbit!

⚠️

Student Fallacy (Why candidates pick this trap): Calculating only the potential energy difference to lift the satellite to altitude 2R without providing orbital speed.

⚠️

NTA Trap Alert: Forgetting Orbital Kinetic Energy

Over 35% of students calculate only ΔU = U(3R) - U(R) = 2GmM/3R (Option 4), forgetting that a satellite in orbit must also have orbital kinetic energy K = GmM/(6R) to keep from falling back down!

🛡️ How to avoid the -1 mark: Always remember: Launch energy = ΔU + K_orbit = E_final - U_initial!

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Core NEET Syllabus Standard

Physics Core Gravitation · Gravitation Standard Benchmark

Core Scientific Principle Tested:

"Total mechanical energy of an orbiting satellite is E = -GmM/(2r); minimum launch energy from surface is E_launch = E_orbit - U_surface."

Syllabus Focus: Energy of an Orbiting Satellite

✓ Core NEET UG Syllabus Anchor Syllabus & Academic Standards ↗

Lift takes 4/6, Orbit takes 1/6: Together they make 5/6!

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NEET 2024 Physics Question 46 Solution

Question

The 30-Second Exam Hall Shortcut: ⚡ Total Energy Balance Shortcut

Step-by-Step Rigorous Workflow & Mathematical Proof

Initial Energy of the Satellite on Earth’s Surface

Orbital Radius Determination

Total Mechanical Energy in Circular Orbit

Calculation of Minimum Launch Energy

360° Option Elimination Engine Key Feature

NTA Trap Alert: Forgetting Orbital Kinetic Energy

Core NEET Syllabus Standard

Ready to attempt under real exam pressure?

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