An artificial satellite revolves around a planet for which gravitational force \((F)\) varies with the distance \(r\) from its centre as \(F \propto r^{2}.\) If \(v_0\) is its orbital speed, then:

1.	\(v_{0} \propto r^{-1/2}\)	2.	\(v_{0} \propto r^{3/2}\)
3.	\(v_{0} \propto r^{-3/2}\)	4.	\(v_{0} \propto r\)

If the gravitational potential on the surface of the earth is V₀, then the potential at a point at a height equal to half of the radius of the earth is:

1. $\frac{{\mathrm{V}}_0}{2}$

2. $\frac{2}{3}{\mathrm{V}}_0$

3. $\frac{{\mathrm{V}}_0}{3}$

4. $\frac{3{\mathrm{V}}_0}{2}$

The total mechanical energy of an object of mass m projected from the surface of the earth with escape speed is:

1. Zero

2. Infinite

3. $-\frac{\mathrm{GMm}}{2\mathrm{R}}$

4. $-\frac{\mathrm{GMm}}{3\mathrm{R}}$

A body is thrown with a velocity equal to n times the escape velocity (v_e). The velocity of the body at a large distance away will be:

1. ${\mathrm{v}}_{\mathrm{e}}\sqrt{{\mathrm{n}}^2-1}$

2. ${\mathrm{v}}_{\mathrm{e}}\sqrt{{\mathrm{n}}^2+1}$

3. ${\mathrm{v}}_{\mathrm{e}}\sqrt{1-{\mathrm{n}}^2}$

4. None of these

The escape velocity of a body from the earth is about 11.2 km/s. Assuming the mass and radius of the earth to be about 81 and 4 times the mass and radius of the moon respectively, the escape velocity in km/s from the surface of the moon will be:

1. 0.54

2. 2.48

3. 11

4. 49.5

If M is the mass of a planet and R is its radius, then in order to become black hole:

[c is speed of light]

1. $\sqrt{\frac{\mathrm{GM}}{\mathrm{R}}}\le \mathrm{c}$

2. $\sqrt{\frac{\mathrm{GM}}{2\mathrm{R}}}\ge \mathrm{c}$

3. $\sqrt{\frac{2\mathrm{GM}}{\mathrm{R}}}\ge \mathrm{c}$

4. $\sqrt{\frac{2\mathrm{GM}}{\mathrm{R}}}\le \mathrm{c}$

The atmosphere on a planet is possible only if: [where v_rms is root mean square speed of gas molecules on planet and v_e is escape speed on its surface]

1. ${\mathrm{v}}_{\mathrm{rms}}={\mathrm{v}}_{\mathrm{e}}$

2. ${\mathrm{v}}_{\mathrm{rms}}>{\mathrm{v}}_{\mathrm{e}}$

3. ${\mathrm{v}}_{\mathrm{rms}}\le {\mathrm{v}}_{\mathrm{e}}$

4. ${\mathrm{v}}_{\mathrm{rms}}<{\mathrm{v}}_{\mathrm{e}}$

When the speed of a satellite is increased by x percentage, it will escape from its orbit where the value of x is:

1. 11.2%

2. 41.4%

3. 27.5%

4. 34.4%

In an orbit if the time of revolution of a satellite is T, then P.E. is proportional to:

1. ${\mathrm{T}}^{1/3}$

2. ${\mathrm{T}}^3$

3. ${\mathrm{T}}^{-2/3}$

4. ${\mathrm{T}}^{-4/3}$

A small satellite is revolving near earth's surface. Its orbital velocity will be nearly:

1. 8 km/s

2. 11.2 km/s

3. 4 km/s

4. 6 km/s