Kepler's laws of planetary motion. These are :
(i) The Law of Orbits: All planets move in elliptical orbits with the sun at one focus.
(ii) The law of Areas: A line that connects a planet to the sun sweeps out equal areas in equal time, i.e., the aerial velocity of the planet is always constant. [This law is based on conversation of angular momentum)
(iii) The Law of Periods: The square of the period of revolution of any planet around the sum is proportional to the cube of the semi-major axis of the orbit, of planet i.e., T2 µ r3.
Velocity given to a satellite in order to keep it in orbit is known as orbital velocity (V0). The necessary centripetal force needed for circular motion is provided by the gravitational attraction between earth of mass Me and satellite of mass m.
The orbital velocity of a mass m at a height ‘h’ above the surface of earth is
If the satellite is very near to the surface of earth, then h = 0
In case of earth
The time taken by a planet in completing one revolution is its time period T. In one revolution it covers a a at speedplinear distance of 2 v. Thus,
or, (Kepler’s Third laws)
The kinetic energy of the planet is
The gravitational potential energy of the sun-planet system is
(where M is the mass of sun and m is the mass of planet)
The total mechanical energy of the sun-planet system is
The total energy is negative. This is true for any bound system if the potential energy is taken to be zero at infinite separation.
Regarding the velocity of projection of a body following cases may be considered:
1.V < V0 Body does not orbit the Earth. It is falls
back on the Earth's surface.
2. Circular path
3. V0 < V < Ve Elliptical path
4. V = Ve Parabolic path
5. V > Ve Hyperbolic path
The net velocity where V0 ® orbital velocity
Ve ® escape velocity
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