ONE OCEAN ACADEMY
Gravitation - Quick Reference Guide
Kepler's Three Laws
1. Law of Orbits: Elliptical orbits, Sun at one focus
Perihelion = closest, Aphelion = farthest
2. Law of Areas: Equal areas in equal times
Consequence of angular momentum conservation
3. Law of Periods: T² ∝ a³
T²/a³ = constant for all planets
Universal Law of Gravitation
F = Gm₁m₂/r²
G = 6.67×10⁻¹¹ Nm²kg⁻²
Force is ALWAYS attractive
If distance doubles → Force becomes 1/4
Acceleration Due to Gravity
At Surface: g = GM/R² ≈ 9.8 m/s²
At Height h: g' = g/(1+h/R)²
Approximation: g' ≈ g(1-2h/R) for h << R
At Depth d: g' = g(1-d/R)
At center (d=R): g = 0
Gravitational Potential Energy
U = -GMm/r
Zero at infinity (convention)
At Surface: U = -GMm/R = -mgR
Approximation: ΔU ≈ mgh
Valid only for h << R
Exact: ΔU = GMmh/R(R+h)
Escape Velocity
vₑ = √(2GM/R) = √(2gR)
For Earth: vₑ ≈ 11.2 km/s
Independent of:
• Mass of body
• Direction of projection
• Height of launch point
Satellites & Orbital Motion
Orbital Velocity: v₀ = √(GM/r)
r = orbital radius from Earth's center
Time Period: T² ∝ r³
Kepler's 3rd law for satellites
Relation: v₀ = vₑ/√2
Energy of Orbiting Satellite
Kinetic Energy: K = GMm/2r
Potential Energy: U = -GMm/r
Total Energy: E = -GMm/2r
E is negative → bound system
Virial Theorem: K = -U/2
Or: |U| = 2K, so E = -K
Important Constants & Data
G
6.67×10⁻¹¹ Nm²kg⁻²
g
9.8 m/s²
M_Earth
6.0×10²⁴ kg
R_Earth
6.4×10⁶ m
M_Sun
2×10³⁰ kg
Earth-Sun
1.5×10¹¹ m
⚡ Quick Problem-Solving Tips
Always check if distance is from center or surface (use R+h for height)
Negative energy = bound system (elliptical orbit)
Zero total energy = parabolic path (just escapes)
Positive energy = hyperbolic path (escapes with leftover speed)
Inside shell: g = 0 | Inside sphere: g ∝ r (linear)
Weightlessness in orbit = free fall (NOT zero gravity!)
g is independent of body mass (only depends on planet M and R)
Essential Formula Sheet
F = Gm₁m₂/r²
g = GM/R²
W = mg
U = -GMm/r
vₑ = √(2gR)
v₀ = √(GM/r)
T² = 4π²r³/GM
E = -GMm/2r
K = GMm/2r
❌ Common Mistakes to Avoid
Wrong: g = GM/h²
Wrong: vₑ depends on mass
Wrong: Zero gravity in orbit
Correct: g = GM/(R+h)²
Correct: vₑ is independent of mass
Correct: Free fall creates weightlessness
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