A body of mass m is situated at a distance 4${\mathrm{R}}_{\mathrm{e}}$ above the Earth's surface, where ${\mathrm{R}}_{\mathrm{e}}$ is the radius of the Earth. What minimum energy should be given to the body so that it may escape? 

1.	mgRe	2.	2mgRe
3.	mgRe/5	4.	mgRe/16

Two satellites S₁ and S₂ are revolving around a planet in coplanar and concentric circular orbits of radii R₁ and R₂ in the same direction respectively. Their respective periods of revolution are 1 hr and 8 hr. The radius of the orbit of satellite S₁ is equal to 10⁴ km. Find the relative speed when they are closest to each other. 

1. $2\mathrm{\pi }\times {10}^4$ $\mathrm{kmph}$

2. $\mathrm{\pi }\times {10}^4$ $\mathrm{kmph}$

3. $\frac{\mathrm{\pi }}{2}\times {10}^4$ $\mathrm{kmph}$

4. $\frac{\mathrm{\pi }}{3}\times {10}^4$ $\mathrm{kmph}$ 

The gravitational potential energy of an isolated system of three particles, each of mass \(\mathrm{m}\) placed at three corners of an equilateral triangle of side \(\mathrm{l}\) is: 

The potential energy of a satellite having mass m and rotating at a height of 6.4 × ${10}^6$m from the Earth's surface is:

1.  $-0.5$ ${\mathrm{mgR}}_{\mathrm{e}}$

2.  $-{\mathrm{mgR}}_{\mathrm{e}}$

3.  $-2$ ${\mathrm{mgR}}_{\mathrm{e}}$

4.  $4{\mathrm{mgR}}_{\mathrm{e}}$

Two particles of mass \(\mathrm{m}\) and \(\mathrm{4m}\) are separated by a distance \(\mathrm{r}.\) Their neutral point is at:

1.  $\frac{\mathrm{r}}{2}$ $\mathrm{from}$ $\mathrm{m}$

2.  $\frac{\mathrm{r}}{3}$ $\mathrm{from}$ $4\mathrm{m}$

3.  $\frac{\mathrm{r}}{3}$ $\mathrm{from}$ $\mathrm{m}$

4.  $\frac{\mathrm{r}}{4}$ $\mathrm{from}$ $4\mathrm{m}$

A satellite is revolving around the earth with speed ${\mathrm{v}}_0$. If it is stopped suddenly, then with what velocity will the satellite hit the ground? (${\mathrm{v}}_{\mathrm{e}}$ = escape velocity from the earth's surface)

1.  $\sqrt{v_e^2 - v_0^2}$

2.  $\sqrt{v_e^2 - 2v_0^2}$

3.  $\sqrt{v_e^2 - 3v_0^2}$

4.  $\sqrt{v_e^2 - \frac{v_0^2}{2}}$

Three identical point masses, each of mass 1 kg lie at three points (0, 0), (0, 0.2 m), (0.2 m, 0). The net gravitational force on the mass at the origin is:
1. \(6.67\times 10^{-9}(\hat i +\hat j)~\text{N}\)
2. \(1.67\times 10^{-9}(\hat i +\hat j) ~\text{N}\)
3. \(1.67\times 10^{-9}(\hat i -\hat j) ~\text{N}\)
4. \(1.67\times 10^{-9}(-\hat i -\hat j) ~\text{N}\)

The figure shows a planet in an elliptical orbit around the sun (S). The ratio of the momentum of the planet at point A to that at point B is:

If R is the radius of the orbit of a planet and T is the time period of the planet, then which of the following graphs correctly shows the motion of a planet revolving around the sun?

1.		2.
3.		4.

If the speed of an artificial satellite revolving around the earth in a circular orbit be \(2 \over 3\) of the escape velocity from the surface of earth then its altitude above the surface of the earth is