A body weights \(63\) N on the surface of the earth. What is the gravitational force on it due to the earth at a height equal to half the radius of the earth?
1. \(98~\text N\) 
2. \(35~\text N\) 
3. \(63~\text N\) 
4. \(28~\text N\) 

Assuming the earth to be a sphere of uniform mass density, how much would a body weigh halfway down to the centre of the earth if it weighed \(250\) N on the surface?

A rocket is fired vertically with a speed of \(5\) km/s from the earth’s surface. How far from the earth does the rocket go before returning to the earth?
1. \(8\times10^6\) m
2. \(1.6\times10^6\) m
3. \(6.4\times10^6\) m
4. \(12\times10^6\) m

The escape speed of a projectile on the Earth’s surface is \(11.2 ~\text{Km/s}.\) A body is projected out with thrice this speed. What is the speed of the body far away from the Earth?  (Ignore the presence of the sun and other planets.)
1. \(32.7 ~\text{Km/s}\)
2. \(11.2 ~\text{Km/s}\)
3. \(31.7 ~\text{Km/s}\)
4. \(21.2 ~\text{Km/s}\)

A satellite orbits the earth at a height of 400 km above the surface. How much energy must be expended to rocket the satellite out of the earth’s gravitational influence? (Mass of the satellite = 200 kg; mass of the earth = 6.0×${10}^{24}$ kg, radius of the earth = 6.4 × ${10}^6$ m, G = 6.67 × ${10}^{-11}$ $N m^2 k{g}^{-2}$)

1. 7.5 × 10⁹ J
2. 4.7 × 10⁹ J
3. 4.4 × 10⁹ J
4. 5.9 × 10⁹ J

Two heavy spheres each of mass \(100~\text{kg}\) and radius \(0.10~\text m,\) are placed \(1.0~\text m\) apart on a horizontal table. What is the gravitational potential at the midpoint of the line joining the centres of the spheres?

A spaceship is stationed on Mars. How much energy must be expended on the spaceship to launch it out of the solar system? (Mass of the space ship = 1000 kg; mass of the sun = 2×${10}^{30}$ kg; mass of mars = 6.4×${10}^{23}$ kg; radius of mars = 3395 km; radius of the orbit of mars = 2.28×${10}^8$ km; G = 6.67×${10}^{-11}$$N m^2 k{g}^{-2}$.)
1. 11 × 10¹⁰ J
2. 6 × 10¹¹ J
3. 6.67 × 10¹⁰ J
4. 7.12 × 10¹¹ J

A rocket is fired vertically from the surface of mars with a speed of 2 km/s. If 20% of its initial energy is lost due to martian atmospheric resistance, how far will the rocket go from the surface of Mars before returning to it?

(Mass of mars = 6.4×${10}^{23}$ kg; radius of mars = 3395 km; G = 6.67×${10}^{-11}$ $N m^2 k{g}^{-2}.$)

1. 477.2 km
2. 550.5 km
3. 493.9 km
4. 517.3 km

According to Kepler, the planets move in:

1. the circular orbits around the sun with the sun at its center.

2. the elliptical orbits around the sun.

3. the straight lines with constant velocity.

4. the elliptical orbits around the sun with the sun at one of its foci.

The minimum and maximum distances of a planet revolving around the sun are \(r\) and \(R.\) If the minimum speed of the planet on its trajectory is \(v_0,\) its maximum speed will be: