Gravitational Field, Potential and Escape Velocity

• Gravitational field intensity: g = GM/r² (field at point, not surface only) — force per unit mass.
• Gravitational potential: V = −GM/r — potential energy per unit mass; negative (bound system).
• Potential energy: U = −GMm/r for mass m at distance r from planet.
• Escape velocity: v_e = √(2GM/R) = √(2gR) ≈ 11.2 km/s for Earth — minimum speed to escape gravity.
• Relationship between field and potential: g = −dV/dr.

Level 1 — Foundations

Verbal: The gravitational field at a point tells you the force per unit mass a test body would feel there. The potential tells you the potential energy per unit mass.

Gravitational field intensity: g = F/m = GM/r² (vector, directed toward mass M).

Gravitational potential: V = −GM/r (scalar, always negative for attractive field). At infinity: V → 0 (reference). Closer to mass: V is more negative (bound more tightly).

Potential energy: U = mV = −GMm/r. Work done to bring mass m from ∞ to r = −GMm/r.

Escape velocity derivation: Set total energy = 0 (just reaches infinity with v = 0): ½mv_e² − GMm/R = 0 → v_e = √(2GM/R) = √(2gR). For Earth: v_e = √(2 × 9.8 × 6.4×10⁶) ≈ 11.2 km/s.

Binding energy: Energy needed to take mass from surface to infinity = GMm/R = ½mv_e².

Level 2 — JEE / NEET depth

Field-potential relation: g_vec = −∇V; in 1D: g = −dV/dr.

Inside uniform sphere (r < R): g_inside = GMr/R³ (increases linearly from 0 at centre to g_surface at r = R). V_inside = −GM(3R² − r²)/(2R³) (parabolic variation, minimum at centre).

Outside sphere (r ≥ R): g = GM/r² (inverse square); V = −GM/r.

On surface (r = R): g_surface = GM/R²; V_surface = −GM/R.

Gravitational potential due to shell:

• Outside: V = −GM/r (same as point mass).
• Inside shell: V = −GM/R (constant! — field = 0 inside shell).

At height h: g_h = GM/(R+h)²; V_h = −GM/(R+h).

Escape velocity independence: v_e doesn't depend on mass of projectile — same for any object.

Escape velocity and orbital velocity relationship: v_e = √2 × v_orbital (at same radius).

Potential and field at various distances

For Earth: M = 6×10²⁴ kg, R = 6.4×10⁶ m, G = 6.67×10⁻¹¹.
Find g and V: (a) at surface, (b) at r = 2R from centre.

Part (a) — At surface (r = R):
g = GM/R² = (6.67×10⁻¹¹ × 6×10²⁴) / (6.4×10⁶)²
  = (4.002×10¹⁴) / (4.096×10¹³) ≈ 9.77 m/s² ≈ 9.8 m/s²

V = −GM/R = −4.002×10¹⁴ / 6.4×10⁶ ≈ −6.25×10⁷ J/kg

Part (b) — At r = 2R:
g = GM/(2R)² = g_surface / 4 ≈ 9.8/4 = 2.45 m/s²

V = −GM/(2R) = V_surface / 2 = −6.25×10⁷ / 2 = −3.125×10⁷ J/kg

Note: g reduced by factor 4 (inverse square); V reduced by factor 2 (inverse r).

Escape velocity comparison

Find escape velocity from Moon.
M_moon = 7.35×10²² kg, R_moon = 1.74×10⁶ m.

Step 1: v_e = √(2GM/R)
       = √(2 × 6.67×10⁻¹¹ × 7.35×10²² / 1.74×10⁶)
       = √(2 × 4.90×10¹² / 1.74×10⁶)
       = √(5.63×10⁶) ≈ 2374 m/s ≈ 2.4 km/s

Step 2 — Compare to Earth (11.2 km/s):
v_e,moon / v_e,earth = √(M_m R_e / M_e R_m) = √(7.35×10²²/6×10²⁴ × 6.4×10⁶/1.74×10⁶)
                      = √(0.01225 × 3.678) = √0.0450 ≈ 0.212 → ~2.37 km/s ✓

Conclusion: Moon's low escape velocity explains why it has no atmosphere.

• Find V at r = 3R from Earth's centre (use V_surface = −6.25×10⁷ J/kg).
• What is the gravitational field inside a uniform spherical shell?
• Show that g reduces by 36% when going from Earth's surface to height h = R/4.
• Why is gravitational potential always negative?
• Stretch: Find the energy needed to move a 500 kg satellite from Earth's surface to r = 2R.

NCERT Chapter 8 link: Gravitational potential is the energy map of the gravitational field — negative values mean the object is bound. Escape velocity follows from setting total mechanical energy to zero. The field inside shells being zero is a key result used in satellite and planetary physics.

Exam connections: JEE tests: g inside/outside Earth (which formula when); gravitational potential difference (work = mΔV); escape velocity calculation and ratio problems; comparing v_e with orbital v. Common trap: using g = GM/r² inside Earth (only valid outside).

Study strategy: Draw a graph of g vs r and V vs r for a uniform sphere — the kink at r = R is a JEE favourite. Inside: g linear; outside: g ∝ 1/r². Inside: V parabolic; outside: V ∝ 1/r.

• Use the platform-native live simulation or PhET-style tool for this topic (number line, Venn, physics playground, molecule builder, sensor dashboard, etc.).
• Mirror / body / home activity: physically do the concept (count objects, measure, role-play) and photograph or describe for portfolio.
• Voice or text reflection with AI Mentor: explain the concept to a younger student or family member.

• "Explain this concept to a Class 6 student using one real example from an Indian home, school, market, or festival."
• "What is one common mistake students make here, and how would you catch yourself making it?"
• Stretch: "How does this connect to coding, robotics, money, health, environment, or a future career?"

• Complete the core practice + one extension activity (photo, table, short reflection, or mini-project) for base XP + topic badge.
• 5-7 day streak or family discussion note = multiplier + visible artifact in parent/principal dashboard.
• Best real-world application stories (anonymised) featured on class or national leaderboard.

• One hands-on project or measurement using the Drishti kit or household items that makes the concept physical.
• Direct link to at least one Future Skill track (Money Management, Green Tech, Cyber Defenders, Micro-Entrepreneurship, AI Mastery, Sustainable Living, Personality Development).
• Coding extension where relevant (simple script, simulation, or data logging).

This material emphasises experiential "learning by doing", competency (apply/create/analyse), vocational exposure, critical thinking, and multidisciplinary connections. Designed to feed live worlds, AI Mentor (with memory), gamification, robotics, parent analytics, and future skills — not just exam prep.

Portfolio Evidence Idea: Your photo/table/reflection/project + one sentence on "How this helps me in real life or a possible future path."

Open the Practice tab for aligned questions (easy/medium/hard + case-based) with full AI scaffolding.

See curriculum for cross-links and the full future-skills/robotics chapters.