Q. No.A particle of mass \(m\) is thrown upwards from the surface of the earth, with a velocity \(u.\) The mass and the radius of the earth are, respectively, \(M\) and \(R.\) \(G\) is the gravitational constant and \(g\) is the acceleration due to gravity on the surface of the earth. The minimum value of \(u\) so that the particle does not return back to earth is:
1. \(\sqrt{\dfrac{2 {GM}}{{R}^2}} \)
2. \(\sqrt{\dfrac{2 {GM}}{{R}}} \)
3.\(\sqrt{\dfrac{2 {gM}}{{R}^2}} \)
4. \(\sqrt{ {2gR^2}}\)
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| 1. | \(11.2 ~\cos60^\circ\) km/s | 2. | \(11.2 ~\sin60^\circ\) km/s |
| 3. | \(11.2\) km/s | 4. | \(11.2 ~\tan60^\circ\) km/s |
If the radius of a planet is \(R\) and its density is \(\rho, \) the escape velocity from its surface will be
1. \(V_{e} \propto p R\)
2. \(V_{e} \propto R \sqrt{\rho}\)
3. \(V_{e} \propto \dfrac{\sqrt{p}}{R}\)
4. \(V_{e} \propto \dfrac{1}{\sqrt{p} R}\)
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| 1. | \(\dfrac{v}{3}\) | 2. | \(\dfrac{2v}{3}\) |
| 3. | \(\dfrac{3v}{4}\) | 4. | \(\dfrac{9v}{4}\) |
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(\(v_{e}\)is escape speed from the Earth.)
1. \(\dfrac{v_e}{\sqrt2}\)
2. \(\dfrac{v_e}{2\sqrt2}\)
3. \(\dfrac{3v_e}{\sqrt2}\)
4. \(\dfrac{v_e}{2}\)
A body is moving in a low circular orbit about a planet of mass \(M\) and radius \(R. \) The radius of the orbit can be taken to be \(R\) itself. The ratio of the speed of this body in the orbit to the escape velocity from the planet is:
1. \(\sqrt{2}\)
2. \(\dfrac{1}{\sqrt{2}}\)
3. \(2\)
4. \(1\)
1. \(2\sqrt{gR} \)
2. \(\left({\sqrt{2}-1}\right)\sqrt{gR} \)
3. \(\dfrac{gR}{2}\)
4. \(\left({\sqrt{3}-1}\right)\sqrt{gR}\)
| 1. | does not depend on the mass of the rocket. |
| 2. | does not depend on the mass of the earth. |
| 3. | depends on the mass of the rocket. |
| 4. | depends on the mass of the planet towards which it is moving. |
| 1. | \(1:2\) | 2. | \(1:1\) |
| 3. | \(2:1\) | 4. | \(3:1\) |
| 1. | \(\sqrt{5gR}\) | 2. | \(\sqrt{3gR}\) |
| 3. | \(\sqrt{2gR}\) | 4. | infinite |
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