Gravitation and Satellites Questions for AGNIVEER VAYU

Newton's Law of Universal Gravitation

Every mass attracts every other mass. The force depends on how massive the objects are and how far apart they sit. Think of gravity as an invisible rope connecting all objects — the heavier the objects, the stronger the rope; the farther apart, the weaker it becomes. This is the foundation for understanding planetary motion and satellite behavior.

• Force is directly proportional to product of masses (M and m)
• Force is inversely proportional to square of distance (r squared)
• Gravitational constant G = 6.67 × 10^-11 N⋅m²/kg² — always given in exam
• Works for point masses and spherically symmetric bodies
• Gravity is always attractive, never repulsive

Orbital Velocity and Circular Orbits

A satellite orbits when gravitational force provides exactly the centripetal force needed to keep it in circular motion. If velocity is too low, it falls; too high, it escapes. The orbital velocity is independent of the satellite's mass — a feather and a boulder orbit at the same speed if at the same altitude. This is counterintuitive but critical for Agniveer questions.

• Gravitational force = Centripetal force: GMm/r² = mv²/r
• Orbital velocity v_o = sqrt(GM/r) — depends only on central mass and radius
• For Earth orbit at surface: v_o ≈ 7.9 km/s (first cosmic velocity)
• Orbital velocity decreases with altitude: higher orbit = slower speed
• Period of orbit T = 2πr / v_o = 2π sqrt(r³/GM)

Escape Velocity

Escape velocity is the minimum speed needed to break free from a planet's gravity and never return. Unlike orbital velocity (which keeps you in circular motion), escape velocity launches you away forever. It's independent of direction — you can escape upward, sideways, or at any angle. For Earth, it's about 11.2 km/s.

• Escape velocity v_e = sqrt(2GM/R) — exactly sqrt(2) times orbital velocity at surface
• For Earth: v_e ≈ 11.2 km/s (second cosmic velocity)
• Independent of mass of escaping object — a rocket and a pebble escape at same speed
• Derived from energy conservation: KE at surface = PE at infinity
• Escape velocity increases with planet mass, decreases with planet radius

Gravitational Potential Energy

Gravitational PE is the energy stored in the gravitational field. Unlike everyday potential energy (which is zero at ground level), gravitational PE is zero at infinity and negative everywhere else. This negative sign is crucial — it means objects are bound to the planet. The more negative the PE, the deeper in the gravity well you are.

• PE = -GMm/r (always negative for bound objects)
• At Earth's surface: PE ≈ -GMm/R
• Total mechanical energy of orbiting satellite: E = KE + PE = -GMm/2r (negative = bound)
• To escape: total energy must be zero or positive
• Potential energy increases (becomes less negative) as you move away from planet

Kepler's Laws and Satellite Motion

Kepler's laws describe how planets and satellites move. The third law is most useful for exams — it relates orbital period to orbital radius. All satellites follow these laws regardless of their mass. These laws emerge naturally from Newton's gravity law but are stated separately because they're so powerful for predictions.

• Kepler 1st Law: Orbits are ellipses with central body at one focus
• Kepler 2nd Law: Equal areas swept in equal times (angular momentum conservation)
• Kepler 3rd Law: T² ∝ r³ or T² = (4π²/GM) × r³
• For circular orbits, r is orbital radius; for ellipses, use semi-major axis
• Useful for comparing satellites: T₁²/T₂² = r₁³/r₂³

• Forgetting that orbital velocity is independent of satellite mass — many students think heavier satellites need higher speeds. Wrong. Speed depends only on the central body and altitude.
• Confusing escape velocity with orbital velocity — escape velocity is sqrt(2) times larger and launches you away forever, not into orbit.
• Using r as Earth's radius instead of orbital radius — always measure from center of Earth. If altitude is h, then r = R + h.
• Treating gravitational PE as positive — it's always negative for bound objects. This trips up energy conservation problems.
• Forgetting that Kepler's 3rd Law uses semi-major axis for ellipses, not just any radius — circular orbits use the radius directly.
• Mixing up 'g' at surface with 'g' at altitude — g decreases as 1/r², so at 2R from center, g becomes g/4, not g/2.

• GOVE: Gravitation, Orbit, Velocity, Escape — the four pillars of satellite physics. Master each separately, then combine.
• Orbital velocity = sqrt(GM/r). Escape velocity = sqrt(2GM/r). Escape is sqrt(2) times orbital. Remember: 1.414 × orbital = escape.
• PE is negative, KE is positive, total energy of orbit is negative (bound). Negative total = trapped. Zero or positive = free.
• Kepler 3: T² goes with r³. If radius doubles, period increases by 2^(3/2) ≈ 2.83 times, not 2 times. Cube root relationship is key.
• ISS memory peg: ~7.8 km/s orbital velocity, ~90 min period, ~400 km altitude. Use this as a reference point for all Earth orbit problems.

• Agniveer Vayu typically asks 1-2 direct calculation questions (orbital velocity, escape velocity, period) and 1-2 conceptual questions (why satellites don't fall, how geostationary orbits work). Practice both types.
• Recent papers favor questions combining two concepts: e.g., 'A satellite is launched from Earth's surface with escape velocity. How high does it go?' (Requires energy conservation + escape velocity knowledge).
• Diagram-based questions are common — you may see orbital diagrams and be asked to identify which satellite is fastest, has longest period, or requires most energy to launch. Faster orbits are lower; slower orbits are higher.
• Time management: Numerical problems take 3-4 minutes. Conceptual questions take 1-2 minutes. Allocate accordingly in your 2-hour exam window.
• Always state your assumptions clearly (e.g., 'assuming circular orbit,' 'neglecting air resistance'). Agniveer examiners value reasoning, not just final answers.
• Use g = 10 m/s² and GM = 4 × 10^14 m³/s² for Earth unless told otherwise — this speeds up calculations significantly.