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Satellite questions look like they need a dozen formulas. They need two ideas: gravity provides the centripetal force for a circular orbit, and energy is conserved when only gravity acts. Most lost marks come from using a near-surface formula far from the surface, or from beliefs about gravity in space that are simply wrong. This post works through orbital speed, satellite energy and escape speed with numbers, and lists the errors to avoid.
What misconceptions about orbits are common?
A review of research on how people understand gravity, Kavanagh and Sneider (2006), found a set of misconceptions "surprisingly widespread, even among university students and teachers". They include the beliefs that there is no gravity in space, that objects in orbit are weightless so gravity does not act on them, that the force of gravity falls off rapidly with altitude, and that a force is needed to keep an object moving along its orbit.
Each one leads to a wrong equation. An astronaut in orbit feels weightless because she and her spacecraft fall together, not because gravity has switched off. At the height of a typical low orbit, gravity is still close to its surface value. And no forward push is needed: gravity alone bends the path into a circle.
How do you find orbital speed and period?
The NCERT Class 11 chapter Gravitation sets gravity equal to the centripetal force for a satellite of mass m at distance r from the centre of the Earth:
GMm/r² = mv²/r, so v = √(GM/r).
Using GM = gR², where R is Earth's radius, this becomes v = √(gR²/r). The period is T = 2πr/v. Notice that m cancels: every satellite at the same radius moves at the same speed, whatever its mass.
What energy does a satellite have?
Its kinetic energy is ½mv² = GMm/2r and its potential energy is −GMm/r. The total is E = −GMm/2r. NCERT points out that this total is negative, which means the satellite is bound to the Earth. A satellite in a larger orbit has more (less negative) total energy but less kinetic energy, so it moves more slowly.
Worked example: a satellite at height R
A 1000 kg satellite orbits at a height equal to Earth's radius, so r = 2R. Take R = 6.4 × 10⁶ m and g = 9.8 m/s².
Orbital speed. v = √(gR²/2R) = √(gR/2) = √(9.8 × 6.4 × 10⁶ / 2) = √(3.14 × 10⁷) ≈ 5.6 × 10³ m/s, or 5.6 km/s.
Period. T = 2π(1.28 × 10⁷)/5600 ≈ 1.44 × 10⁴ s, about 4.0 hours.
Gravity at that height. g′ = g(R/r)² = g/4 ≈ 2.45 m/s².
Energy to put it there from rest on the surface (ignoring Earth's rotation). Initial energy is −GMm/R = −mgR. Final energy is −GMm/4R = −mgR/4. The energy needed is ¾mgR = 0.75 × 1000 × 9.8 × 6.4 × 10⁶ ≈ 4.7 × 10¹⁰ J.
How is escape speed calculated, and what does it depend on?
Escape speed is the launch speed for which the total energy is exactly zero, so the body can just reach infinity: ½mv² − GMm/R = 0, giving v = √(2GM/R) = √(2gR). With the numbers above, √(2 × 9.8 × 6.4 × 10⁶) ≈ 11.2 km/s, the value NCERT quotes. NASA's Earth fact sheet lists 11.186 km/s.
Escape speed does not depend on the mass of the body or on the direction of launch, as long as it does not hit the Earth. From an orbit of radius r, escape speed is √(2GM/r), which is √2 times the orbital speed there. For our satellite, √2 × 5.6 ≈ 7.9 km/s.
Which mistakes come up most often?
Using mgh at large heights. As the callout shows, the error can be a factor of two.
Using r = h. In every orbital formula, r is the distance from Earth's centre, R + h.
Thinking a heavier satellite needs more speed. Orbital speed and escape speed are both independent of the satellite's mass. The energy needed does depend on mass.
Mixing up orbital and escape speed. They differ by a factor of √2 at the same radius.
Forgetting the sign of energy. Bound orbits have negative total energy. If yours is positive, the satellite has escaped or there is a sign error.
These ideas link to oscillations, too: a body dropped through a tunnel through a uniform Earth has the same period as a satellite skimming the surface, as our post on showing that a system is SHM works out. For the wider pattern of mechanics errors, see common mistakes in JEE mechanics.
If the energy method itself is the gap, the Fermi Concept Library covers every concept in physics with its own lesson, from gravitational potential to circular motion, so you can fill that one gap before returning to problems.
A practice routine
Day 1: Derive v, T and E for a circular orbit from scratch, without notes.
Day 2: Four numerical orbit problems at different heights, writing r = R + h each time.
Day 3: Three energy problems: launch to orbit, orbit to higher orbit, and escape from orbit.
Day 4: Variation of g with height and depth, then a timed mixed set.
Tonight, redo one satellite question where you used mgh, and solve it again with −GMm/r.
