A satellite orbits the earth at a height of 400 km above the surface. How much energy must be expended to rocket the satellite out of the earth’s gravitational influence? Mass of the satellite = 200 kg; mass of the earth = 6.0 ×10²⁴ kg; radius of the earth = 6.4 ×10⁶ m; G = 6.67 × 10^–11 N m² kg^–2.

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Solution

Mass of the Earth, M = 6.0 × 10²⁴ kg

Mass of the satellite, m = 200 kg

Radius of the Earth, R_e = 6.4 × 10⁶ m

Universal gravitational constant, G = 6.67 × 10^–11 Nm²kg^–2

Height of the satellite, h = 400 km = 4 × 10⁵ m = 0.4 ×10⁶ m

Total energy of the satellite at height h

Orbital velocity of the satellite, v=

Total energy of height, h

The negative sign indicates that the satellite is bound to the Earth. This is called bound energy of the satellite.

Energy required to send the satellite out of its orbit = – (Bound energy)

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