Two bodies of masses ${\mathrm{m}}_1$ and ${\mathrm{m}}_2$  are initially at rest at infinite distance apart. They are then allowed to move towards each other under mutual gravitational attraction. Their relative velocity of approach at a separation distance r between them is

(1) ${\left[2\mathrm{G}\frac{({\mathrm{m}}_1-{\mathrm{m}}_2)}{\mathrm{r}}\right]}^{1/2}$                         

(2) ${\left[\frac{2\mathrm{G}}{\mathrm{r}}({\mathrm{m}}_1+{\mathrm{m}}_2)\right]}^{1/2}$

(3) ${\left[\frac{\mathrm{r}}{2\mathrm{G}({\mathrm{m}}_1{\mathrm{m}}_{2)}}\right]}^{1/2}$                             

(4) ${\left[\frac{2\mathrm{G}}{\mathrm{r}}{\mathrm{m}}_1{\mathrm{m}}_2\right]}^{1/2}$

The rotation period of an earth satellite close to the surface of the earth is 83 minutes. The time period of another earth satellite in an orbit at a distance of three earth radii from its surface will be

(1) 83 minutes                                   

(2) $83\times \sqrt{8}$ minutes

(3) 664 minutes                                 

(4) 249 minutes

A satellite of mass m is circulating around the earth with constant angular velocity. If the radius of the orbit is $R_0$ and mass of the earth M, the angular momentum about the centre of the earth is:

(1)$m\sqrt{GM{R}_0}$                                         

(2)$M\sqrt{Gm{R}_0}$

(3)$m\sqrt{\frac{GM}{R_0}}$                                           

(4)$M\sqrt{\frac{GM}{R_0}}$

If the distance between the earth and the sun becomes half its present value, the number of days in a year would have been
(1) 64.5                                     

(2) 129

(3) 182.5                                   

(4) 730

A geostationary satellite orbits around the earth in a circular orbit of radius 36000 km. Then, the time period of a satellite orbiting a few hundred kilometres above the earth’s surface (${\mathrm{R}}_{\mathrm{earth}}$= 6400 km) will approximately be -

(1) 1/2 h                   

(2) 1 h

(3) 2 h                       

(4) 4 h

A planet revolves around the sun whose mean distance is 1.588 times the mean distance between earth and the sun. The revolution time of the planet will be:

(1) 1.25 years                               

(2) 1.59 years

(3) 0.89 years                               

(4) 2 years

The earth E moves in an elliptical orbit with the sun S at one of the foci as shown in figure. Its speed of motion will be maximum at the point

(1) C

(2) A

(3) B

(4) D

The period of revolution of planet A around the sun is 8 times that of B. The distance of A from the sun is how many times greater than that of B from the sun ?

(1) 2                                         

(2) 3                                      

(3) 4                                        

(4) 5

If the radius of earth's orbit is made 1/4 th, the duration of an year will become -

(1) 8 times                                     

(2) 4 times

(3) 1/8 times                                   

(4) 1/4 times

If the mass of a satellite is doubled and the time period remain constant, the ratio of orbit in the two cases will be

(1)   1 : 2                                   

(2)   1 : 1

(3)   1 : 3                                   

(4)   None of these