A planet revolves about the sun in elliptical orbit of semi-major axis 2 × 10 12 m . The areal velocity of the planet when it is nearest to the sun is 4 .4 × 10 16 m 2 / s .The least distance between planet and the sun is 1 .8 × 10 12 m . The minimum speed of the planet in km / s is 10 K . Value of K is

Step-by-step solution

Area covered by line joining planet and sun in time dt is ds = 1 2 x 2 d θ ; Areal velocity = ds dt = 1 2 x 2 d θ / dt = 1 2 x 2 ω , where x = distance between planet and sun ω = angular speed of planet about the sun. From kepler's second low, areal velocity of planet is constant. At farthest position A = ds dt = 1 2 2R - r 2 ω = 1 2 2R - r V B A = 1 2 2R - r V B or V B = 2A 2R - r (least speed). (Using values) V B = 40 km/s Thus, K = 4