Q1. Find the escape velocity from the Moon (g = 1.6 m/s², R = 1.74×10⁶ m).
v_e = √(2gR)
v_e = √(2 × 1.6 × 1.74×10⁶) = √(5.57×10⁶)
v_e ≈ 2360 m/s ≈ 2.36 km/s
Gravitation — Class 11
The same law rules falling apples and orbiting satellites. Compute orbital and escape velocity and master Kepler's laws.
Universal law of gravitationF = G m₁ m₂ / r², where G = 6.67×10⁻¹¹ N·m²/kg². It is the force behind orbits and tides.
Variation of gg = GM/R². It decreases with altitude and depth, and varies slightly with latitude due to Earth's rotation and shape.
Orbital & escape velocityOrbital velocity v₀ = √(GM/r) keeps a satellite in orbit; escape velocity v_e = √(2gR) = √2 × v₀ lets it leave for good.
Kepler's lawsOrbits are ellipses (1st); equal areas are swept in equal times (2nd); T² ∝ r³ (3rd).
Escape & orbital velocity calculator Interactive
Enter a planet's surface gravity and radius to get the escape and orbital velocities.
Escape velocity, √(2gR)—m/s
Orbital velocity, √(gR)—m/s
For Earth (g = 9.8, R = 6.4×10⁶ m) the escape velocity is about 11.2 km/s — the speed a rocket must reach to leave.
Numericals practice Interactive
Escape-velocity and gravitation problems with steps.
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Solved numericals
Q2. Two satellites orbit at radii in the ratio 1:4. Find the ratio of their periods.
Kepler's 3rd law: T² ∝ r³
T₁²/T₂² = (1/4)³ = 1/64
T₁/T₂ = 1/8
Formula sheet
| Quantity | Formula | SI unit |
|---|---|---|
| Universal law | F = G m₁ m₂ / r² | N |
| Acceleration due to gravity | g = G M / R² | m/s² |
| Orbital velocity | v₀ = √(g R) (near surface) | m/s |
| Escape velocity | v_e = √(2 g R) | m/s |
| Kepler's third law | T² ∝ r³ | — |
Common mistakes & exam wins
- Escape velocity v_e = √2 × orbital velocity, and does not depend on the mass of the escaping body.
- g decreases both as you go UP (altitude) and DOWN (depth) from the surface.
- Kepler's 3rd law T² ∝ r³ links a planet's period to its orbit size.
- A geostationary satellite has a period of 24 h and orbits ~36,000 km above the equator.
Frequently asked questions
What is escape velocity?
The minimum speed a body needs to escape a planet's gravity without further propulsion, v_e = √(2gR) ≈ 11.2 km/s for Earth.
What is orbital velocity?
The speed needed to keep a satellite in a stable orbit, v₀ = √(GM/r); near the surface v₀ = √(gR).
What is Kepler's third law?
The square of a planet's orbital period is proportional to the cube of its mean orbital radius: T² ∝ r³.