Find gravitational field at a distance of $$2000 \mathrm{~km}$$ from the centre of earth. (Given $$R_{\text {earth }}=6400 \mathrm{~km}, r=2000 \mathrm{~km} \text {, } M_{\text {earth }}=6 \times 10^{24} \mathrm{~kg} \text { ) }$$

Two satellites $$A$$ and $$B$$ revolve round the same planet in coplanar circular orbits lying in the same plane. Their periods of revolutions are $$1 \mathrm{~h}$$ and $$8 \mathrm{~h}$$, respectively. The radius of the orbit of $$A$$ is $$10^4 \mathrm{~km}$$. The speed of $$B$$ is relative to $$A$$. When they are closed in $$\mathrm{km} / \mathrm{h}$$ is

A planet is revolving around the sun in a circular orbit with a radius $$r$$. The time period is $$T$$. If the force between the planet and star is proportional to $$r^{-3 / 2}$$, then the square of time period is proportional to

The weight of a body on the surface of the earth is 63 N. What is the gravitational force on it due to the earth at a height equal to half the radius of the earth?

A space ship is launched into a circular orbit close to earth’s surface. What additional velocity has now to be imparted to the spaceship in the orbit to overcome the gravitational pull?

(Radius of earth = 6400 km, g = 9.8 m/s$$^2$$)

What is the maximum height attained by a body projected with a velocity equal to one-third of the escape velocity from the surface of the earth? (Radius of the earth $$=R$$ )

Two satellites $$S_1$$ and $$S_2$$ are revolving round a planet in coplanar circular orbits of radii $$r_1$$ and $$r_2$$ in the same direction, respectively. Their respective periods of revolution are $$1 \mathrm{~h}$$ and $$8 \mathrm{~h}$$. The radius of orbit of satellite $$S_1$$ is equal to $$10^4 \mathrm{~km}$$. What will be their relative speed (in $$\mathrm{km} / \mathrm{h}$$) when they are closest?