Q.Write the three laws of Kepler about the motion of planets. OR If a particle is thrown with two times the escape velocity from the surface of the Earth, what would be the remaining velocity at infinity of this particle?

Kepler's first law fixes the shape/geometry of planetary orbits; his second law is a statement of conservation of angular momentum.

Law of orbits (Kepler's first law): Every planet revolves around the Sun in an elliptical orbit, with the Sun situated at one of the two foci of the ellipse (not at the centre). The degree of ellipticity varies from planet to planet, but the Sun always occupies a focus.

Law of equal areas (Kepler's second law): The line segment (radius vector) joining the Sun to a planet sweeps out equal areas in equal intervals of time, i.e. the areal velocity $\dfrac{dA}{dt}$ of the planet is constant throughout its orbit. As a consequence, the planet moves fastest when it is closest to the Sun (perihelion) and slowest when farthest (aphelion). This law is a direct consequence of the conservation of angular momentum, since the gravitational force on the planet acts along the radius vector (a central force), so it exerts zero torqu …

Kepler's three laws: (1) elliptical orbits with the Sun at a focus, (2) equal areas swept in equal times, (3) T^2 proportional to a^3.

1. Law of Orbits: Every planet revolves around the Sun in an elliptical orbit, with the Sun located at one of the two foci of the ellipse (not at the centre).

2. Law of Areas: The line segment joining a planet to the Sun (the radius vector) sweeps out equal areas in equal intervals of time. This means the planet moves faster when it is closer to the Sun (near perihelion) and slower when farther away (near aphelion). This law is a direct consequence of the conservation of angular momentum, since the Sun's gravitational force on the planet is a central force (always along the line joining them), producing zero torque about the Sun.

   …