Simulation · Physics · Class 11

Launch speed: fall, orbit or escape

From the lesson Earth satellites in Gravitation. Change the values and watch what happens.

The idea behind it

NCERT §7.9

A satellite is a body that revolves around the earth; Kepler's laws apply to it just as to planets around the sun. The Moon is the earth's only natural satellite, with a period of about 27.3 days.
For a circular orbit of radius R_E + h, gravity supplies the centripetal force: m v²/(R_E + h) = G M_E m / (R_E + h)², so v = √(G M_E / (R_E + h)), less for higher orbits.
Just above the surface (h ≈ 0) the orbital speed is v = √(g R_E) ≈ 7.9 km/s, and the escape speed is √2 times this.
Period: T = 2π (R_E + h)^(3/2) / √(G M_E), so T² = k (R_E + h)³ with k = 4π²/(G M_E). This is Kepler's third law for satellites.
For an orbit just above the surface, T₀ = 2π √(R_E/g) ≈ 85 minutes.
The same law weighs other planets. Phobos circles Mars in 7 h 39 min at 9.4 × 10³ km, which gives a Mars mass of about 6.48 × 10²³ kg; with its orbit 1.52 times the earth's, a Martian year is (1.52)^(3/2) × 365 ≈ 684 days.
An astronaut in orbit feels weightless not because gravity is small there, but because the astronaut and the craft are falling freely towards the earth together.

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