State and prove Kepler's second law (Law of Areas) of planetary motion.

Explanation

`implies` ..The line that joins any planet to the sun sweeps equal areas in equal intervals of time... It is shown in figure.
`implies` The planet P moves around the sun in an elliptical orbit. The shaded area is the area `DeltaA` swept out in a small interval of time `Deltat` .
`implies` This law comes from the observations that planets appear to move slower when they are farther from the sun than when they are nearer.
`implies` The law of areas can be understood as a consequence of conservation of angular momentum which is valid for any central force.
`implies` The line of action of force which is acting on planet is passing through the sun, hence the torque `tau = r F sin pi = 0` . So the conservation of angular momentum of any planet remains constant.
`implies` Let the sun be at the origin and momentum of the planet be `vecr and vecp` .
`implies` The area swept out by the planet of mass m in time interval `Deltat` is `DeltaA`.
`implies` From figure area of right triangle `DeltaSPP.`
(Note : For `Deltat rarr 0`)
`DeltaA = 1/2 xx ` base ` xx` perpendicular
`DeltaA =1/2 (vecr xx vecv Deltat) " "...(1)`
(Here , `"PP". = DeltaS = vDeltat` )
`implies` Dividing `Deltat` on both side of equation (1) ,
`(DeltavecA)/(Deltat)=1/2(vecrxxvecv)`
but momentum of planet `vecp = mvecv`
`:. vecv = vecp/m`
`:. (DeltavecA)/(Deltat)=1/2(vecrxxvecp/m)`
`:. (DeltavecA)/(Deltat)=1/(2m) (vecrxxvecp)`
`:. (DeltavecA)/(Deltat)=vecL/(2m)" "...(2)`
`( :. vecr xx vecp = vecL` Angular momentum )
`implies` Where  `(DeltavecA)/(Deltat)` is velocity, `vecL` is angular momentum.
`implies` The gravitation force of sun on planet `vecF` is opposite to `vecr` and hence `tau = 0` and L = constant
`implies` So, during revolution of planet remains constant.
[ `:.` m is mass which remains constant]
`:. (DeltavecA)/(Deltat)` = constant `" ".....(3)`
`implies` At This is Kepler law of areas for planetary motion.