A particle of mass \(m\) is projected with a velocity, \(v=kV_{e} ~(k<1)\) from the surface of the earth. The maximum height, above the surface, reached by the particle is: (Where \(V_e=\) escape velocity, \(R=\) radius of the earth)
1. | \(\dfrac{R^{2}k}{1+k}\) | 2. | \(\dfrac{Rk^{2}}{1-k^{2}}\) |
3. | \(R\left ( \dfrac{k}{1-k} \right )^{2}\) | 4. | \(R\left ( \dfrac{k}{1+k} \right )^{2}\) |
The escape velocity from the Earth's surface is \(v\). The escape velocity from the surface of another planet having a radius, four times that of Earth and the same mass density is:
1. | \(3v\) | 2. | \(4v\) |
3. | \(v\) | 4. | \(2v\) |
1. | \({S \over 2},{ \sqrt{3gS} \over 2}\) | 2. | \({S \over 4}, \sqrt{3gS \over 2}\) |
3. | \({S \over 4},{ {3gS} \over 2}\) | 4. | \({S \over 4},{ \sqrt{3gS} \over 3}\) |
1. | \(\dfrac R {n^2}\) | 2. | \(\dfrac {R~(n-1)} n\) |
3. | \(\dfrac {Rn} { (n-1)}\) | 4. | \(\dfrac R n\) |
1. | \(180 ~\text{N/kg}\) | 2. | \(0.05 ~\text{N/kg}\) |
3. | \(50 ~\text{N/kg}\) | 4. | \(20 ~\text{N/kg}\) |
Assuming the earth to be a sphere of uniform density, its acceleration due to gravity acting on a body:
1. | increases with increasing altitude. |
2. | increases with increasing depth. |
3. | is independent of the mass of the earth. |
4. | is independent of the mass of the body. |
Two planets are in a circular orbit of radius \(R\) and \(4R\) about a star. At a specific time, the two planets and the star are in a straight line. If the period of the closest planet is \(T,\) then the star and planets will again be in a straight line after a minimum time:
1. | \(+\frac K2\) | 2. | \(-\frac{K}{2}\) |
3. | \(-\frac{K}{4}\) | 4. | \(+\frac K4\) |