Maharashtra State Board · Class 10 · Physics · Chapter 10

Space Missions — Formula Sheet

Board Formulas

10 formulas

1.Condition for a Circular Orbit
$\frac{m v _{c}^{2}}{R + h} = \frac{GM m}{( R + h ) ^{2}}$
$m$ : Mass of the satellite (kg) · $M$ : Mass of the Earth = 6 × 10²⁴ kg · $R$ : Radius of the Earth = 6.4 × 10⁶ m · $h$ : Height of the satellite above the Earth's surface (m)
The Earth's gravitational force on the satellite provides exactly the centripetal force needed for circular motion at height h.
2.Critical Velocity of a Satellite★
$v_{c} = \frac{GM}{R + h}$
$v_{c}$ : Critical (orbital) velocity (m/s) · $G$ : Universal gravitational constant = 6.67 × 10⁻¹¹ N m²/kg² · $R + h$ : Radius of the orbit, from the Earth's centre (m)
The tangential speed a satellite must have to move in a circular orbit at height h. It does not depend on the satellite's own mass, and it decreases as h increases.
3.Critical Velocity Just Above the Surface
$v_{c} = \frac{GM}{R} = g R \approx 7.9 km/s$
Put h ≈ 0 and use g = GM/R². With g = 9.8 m/s² and R = 6.4 × 10⁶ m this is about 7.9 km/s.
4.Period of Revolution of a Satellite★
$T = \frac{2 π ( R + h )}{v _{c}}$
$T$ : Period of revolution (s)
Time for one revolution = circumference of the orbit ÷ speed. Gives T in seconds when lengths are in m and v_c in m/s.
5.Period in Terms of Orbit Radius
$T = 2 π \frac{( R + h ) ^{3}}{GM}$
Substitute v_c = √(GM/(R + h)) into T = 2π(R + h)/v_c. Higher orbit ⇒ longer period.
6.Kepler's Third Law for Satellites
$T^{2} \propto (R + h)^{3}$
For satellites of the same planet, T²/r³ is the same. If r becomes 4 times, T becomes 4^(3/2) = 8 times.
7.Dependence of Speed on Orbit Radius
$v_{c} \propto \frac{1}{R + h}$
A satellite in a higher orbit moves more slowly. If r becomes 9 times, v_c becomes one-third.
8.Escape Velocity★
$v_{esc} = \frac{2 GM}{R} = 2 g R$
$v_{esc}$ : Escape velocity (m/s) · $R$ : Radius of the planet (m)
Minimum velocity with which an object must be projected from the surface to escape the planet's gravity. For the Earth, about 11.2 km/s.
9.Escape Velocity and Critical Velocity
$v_{esc} = 2 v_{c} (near the surface)$
11.2 km/s ≈ 1.414 × 7.9 km/s. A launch speed between 7.9 and 11.2 km/s cannot make the object escape the Earth.
10.Classification of Satellite Orbits
$LEO: 180 to 2000 km, MEO: 2000 to 35780 km, HEO: h \geq 35780 km$
Heights above the Earth's surface. Low Earth orbit: satellites for scientific experiments and atmospheric studies, the International Space Station and the Hubble telescope (one revolution in about 90 min). Medium: elliptical polar orbits for studying polar regions (period 2 to 24 h), and GPS satellites at about 20,200 km. High: geosynchronous satellites (period about 24 h), used for meteorology and for telephone, TV and radio signals.